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PERSPECTIVE
Cite this: Phys. Chem. Chem. Phys., 2013,
15, 9468
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Vibrational self-consistent field calculations for
spectroscopy of biological molecules: new algorithmic
developments and applications
Tapta Kanchan Roya and R. Benny Gerber*ab
This review describes the vibrational self-consistent field (VSCF) method and its other variants for
computing anharmonic vibrational spectroscopy of biological molecules. The superiority and limitations
of this algorithm are discussed with examples. The spectroscopic accuracy of the VSCF method is
Received 18th February 2013,
Accepted 10th April 2013
compared with experimental results and other available state-of-the-art algorithms for various
DOI: 10.1039/c3cp50739d
of computational effort is investigated. The accuracy of the vibrational spectra of biological molecules
biologically important systems. For large biological molecules with many vibrational modes, the scaling
using the VSCF approach for different electronic structure methods is also assessed. Finally, a few open
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problems and challenges in this field are discussed.
I. Introduction
Vibrational spectroscopy ranks with the most important tools
in the chemical sciences. It is a powerful and widely applicable
approach for characterizing the local and global structures,
bonding and dynamical properties of polyatomic molecules.
The method is ‘‘ever green’’: old as is the history of the field, new
experimental techniques and theoretical tools of interpretation
keep being invented, to cope with novel challenges. A direction
of great current interest, where dramatic recent progress was
made, both theoretically and experimentally, is the spectroscopy
of large molecules and macromolecules, and in particular
biological molecules. Among the important goals of vibrational
spectroscopy for biological molecules is to help in the determination
of their structure, when the other methods can’t be applied; to study
different conformers, and to learn of the underlying potential
surfaces that govern them. There has been substantial development
going on in the area of high resolution spectroscopy, both
experimentally and theoretically, for complex systems like
biological molecules. In the field of experimental spectroscopy,
the efficacious developments in experimental tools produce
extremely accurate spectroscopic information for the measurement
of vibrational spectra of biological molecules. Matrix spectroscopy,1–7
jet expansion techniques,8–13 molecules in super fluid helium
a
Institute of Chemistry and The Fritz Haber Research Center,
The Hebrew University, Jerusalem 91904, Israel
b
Department of Chemistry, University of California, Irvine, CA 92697, USA.
E-mail: [email protected]
9468
Phys. Chem. Chem. Phys., 2013, 15, 9468--9492
droplet methods14–17 are some of them which produce very
reliable high resolution spectroscopic information. Not so
recent, but still a very effective and useful approach for the
determination of infra red (IR) spectra of biological molecules
is matrix spectroscopy. Commonly, inert gas matrices at low
temperature are used as host solids where guest biological
molecules are embedded. The concentration of the sample is
kept very low to ensure that the target molecules are surrounded
only by the inert host. It is expected that the perturbation effects
on the target molecules due to the matrix are relatively small. At
this situation, the isolated molecules are devoid of collisions or
spectral congestion. This leads to sharper spectra with narrow
line widths. This spectral sharpening allows us to study different
conformers of target biological molecules with a considerable
success. However, it is found, as a drawback of this method, that
for many biological molecules the spectra are not sufficiently
well resolved. This is due to the fact that many molecules can
occupy multiple sites in the matrix, and for more flexible and
large molecules this probability is much higher and that causes
severe problems in spectral resolution. Some alternative important
recent developments for the high resolution spectroscopy of
biological molecules are jet-expansion techniques. The main merit
of this method is that the molecules can be isolated precisely at
very low temperature (a few degrees Kelvin) and that results in
high resolution spectroscopic data. Using this technique, one can
get vibrational spectroscopy of the ground as well as the excited
electronic states. Spectroscopic studies of biological molecules in
molecular beams18–31 have drawn much attention in this field.
In these techniques, a molecular beam is crossed at right angles
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along with the output of a laser with stabilized frequency. In
this case perfect isolation of molecules can be attained without
environmental effects. Another new and interesting technique is
the super fluid He droplet method in an ultracold environment
(B0.4 K). Due to the low polarizability of 4He it is expected to have
little environmental effect on immersed species. Additionally,
4
He droplets have a negligible superfluid friction and provide a
homogenous environment at very low temperatures, and these
properties of the system are useful for obtaining high resolution.
Hence, the spectrum should have smaller perturbation than the
noble gas matrix technology. This elegant method has not yet been
explored extensively, but it seems to be very promising for future.
All these novel experimental tools provide very accurate high
resolution spectroscopic data and in coming years one should
expect more dramatic improvements in this field. This invokes
challenges to theoreticians to develop more reliable and efficient
theoretical algorithms to calculate accurate vibrational spectroscopic data for biological molecules.
In the formative years of vibrational spectroscopy, theoretical
treatments of vibrational spectra of polyatomic molecules were
restrained to harmonic oscillator (HO) approximation considering
rigid rotor models of molecular rotation.32 The HO approach was
tested for the calculation of vibrational spectra of biological
molecules.33–35 Such treatment is sometimes useful for rigid
molecules, but has limited accuracy for flexible systems. For
example, most of the biological molecules suffer from floppy
vibrations with strong anharmonic effects. Those are occasionally
involved in various intra and inter molecular hydrogen bonding
and the anharmonic effects are greater for such weakly bound
systems. For example, one of the reasons for the stability of
the DNA double helix is the hydrogen bonds between the
complementary nucleotide base pairs and it is already known
that those bonds are quite soft vibrational modes and are
strongly anharmonic. Formation of several weak hydrogen
bonded complexes with water molecules is another common
feature of biological molecules. The anharmonic effect can also
be observed for some high frequency relatively rigid bonds such
as OH, CH and NH stretching modes. These commonly present
vibrational modes are very important for biological molecules
and can posit anharmonicity as large as 10%. Moreover, the
major experimental data available in the literature for biological
molecules are mostly of these kinds. Due to such considerable
anharmonic effects, investigation of these types of interactions
attracted much interest in this field. Hence, to attain good
accuracy in the calculated vibrational spectra of biological
molecules we need to consider anharmonic treatment.
In the HO approximation, the vibrational Hamiltonian can
be separated into a set of one-dimensional HO using normal
mode coordinates.32 That gives an analytical solution for the
wave functions and energies, which can be calculated very fast
computationally. Though the HO approximation is conceptually
much simpler to use, it is less applicable due to lack of accuracy.
However, the situation is different if one goes beyond the HO
approximation. The main problem of invoking anharmonic
spectroscopy calculations is that different vibrational modes
are not mutually separable, and that makes the anharmonic
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Hamiltonian inherently non-separable. This problem has no
analytical solution since no coordinate system reduces the
anharmonic Hamiltonian to a sum of independent one-dimensional
oscillators. It is basically a quantum many body problem and
rigorous numerical approaches are needed to solve it for many
coupled degrees of freedom. But, this is computationally a far
more demanding task than HO approximation. The CPU time
increases rapidly with the increase in the size of molecule. So,
one needs a suitable algorithm for the treatment of large
anharmonic systems which will balance between accuracy and
computational time. Vibrational self-consistent field (VSCF)
theory is one such approach which has been used extensively
for this purpose. In this review we will discuss mainly the
VSCF approach and its recent developments and extensions
which appear to be an appropriate tool for the anharmonic
spectroscopic calculation of biological molecules. We will,
however, make some comments that may serve to relate VSCF
to other methods in use, or potentially applicable to the system
of interest.
It is well known that most of the vibrational spectroscopy
data obtained using HO approximation never reach experimental
accuracy. There is a considerable deviation from experimental
results to frequencies and intensities from HO approximation,
for all or nearly all fundamental transitions. On the other hand,
high resolution experimental techniques that explore several
systems with high anharmonic effects stimulate the progress
in the anharmonic vibrational spectroscopic calculations. Using
the advanced experimental techniques, vibrational overtone
spectroscopy36,37 of the CH, OH and NH groups in particular,
vibrational spectroscopy of the van der Waals and hydrogen
bonded clusters,38–40 spectroscopy of Intermolecular Vibrational
Energy Redistribution (IVR)41 in the time or the frequency
domain, and many others reveal strong anharmonic effects.
Several algorithmic methods have been developed in parallel
for the computation of vibrational spectroscopy to address the
anharmonic nature of molecular vibrations. There are varieties
of methods available in the literature for the description of
anharmonic vibrational spectroscopy. The empirical scaling
factors approach is one of the widely used methods.42–46 In this
approach, the value of the scaling factors for the theoretical
harmonic vibrational frequencies is typically determined by a
comparison with the corresponding experimental fundamentals,
and least-squares fitting to a set of experimental vibrational
frequencies. It is also found that different scaling factors are
needed for low and high frequency vibrations. Application of an
empirical scaling factor in order to bring computed frequencies
into closer agreement with experimental values has been applied
to compensate for the anharmonicity. This scaling factor is
specific to methods, basis sets and nature of vibrational modes
in a molecule which already have experimental data. For example,
Scott and Radom42 found that the scaling factors for BLYP,
B3LYP and MP2 methods with 6-31G(d) basis are 0.9945,
0.9614 and 0.9427 respectively. In some cases this empirical
treatment to predict anharmonic frequencies from computed
harmonic values reproduced the experimental spectroscopy data
with a good accuracy. However, this procedure is not a first
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principles based approach and does not have a unique solution
if the experimental data are limited for a system. This method
does not construct any underlying potential to represent the
system and, hence, does not deal with the challenge of computing
the other observables from the anharmonic potential energy
surface. This motivates the first principles based calculation of
the realistic anharmonic potential energy surfaces to reproduce
the experimental data.
The most extensively used first principles based method
with great accuracy and moderate computational time for the
calculation of anharmonic spectroscopy of a biological molecule
is the vibrational self-consistent field (VSCF) method and its other
variations. In the late 1970s, the VSCF method was developed by
Bowman,47 Carney et al.,48 Cohen et al.,49 and Gerber and
Ratner.50 Since then it has drawn great attention to several
authors.51–55 Initially VSCF was introduced to apply for a system
of a few coupled anharmonic oscillators. Later, VSCF and its other
improved variants were used for the anharmonic spectroscopy
calculations including macromolecules such as small biological
molecules,56–59 peptides,60 biological molecules in the crystal61–63
and even the BPTI protein64 with considerable accuracy.
One important improved variant of VSCF introduced by Gerber
and co-workers that improves the VSCF energies, keeping the
computational cost in control for relatively large molecules, is
VSCF-PT2 (also referred to as CC-VSCF in the literature), the
second order perturbation corrected VSCF theory.65,66 It was
found that the VSCF-PT2 method is more accurate than VSCF
within the separable approximation, but computationally more
time demanding. Thus further modifications were introduced67,68
that improved the scaling of VSCF-PT2 with the number of
degrees of freedom and made it suitable for biological molecules.
Earlier Gerber and Ratner50 showed that VSCF methods
employed in semi-classical theory are accurate enough but do
not save on computational efforts compared with a quantum
approach. Thus it is preferable to use a fully quantum mechanical
method for most of the systems. However, in some special cases,
where mathematical simplicity is needed, the semi-classical VSCF
can offer greater advantages. For example, as demonstrated by
Gerber and co-workers,69,70 it is advantageous for semi-classical
VSCF to do the direct inversion of vibrational spectroscopic data
in order to obtain the multidimensional potential energy surface
of a system. Due to the separable approximation in VSCF, it
inherently does not consider correlation effects in the modes.
As an algorithmic development, it is desirable to rectify this
by inclusion of the correlation between different modes for
satisfactory agreement with experiments. The inclusion of correlation effects between vibrational modes using a configurational
interaction (CI) method was first introduced by Bowman et al.,71
Ratner et al.,72 and Thompson and Truhlar,73 and applied for small
molecules. This method is highly accurate but computationally very
expensive. The computational time increases exponentially with the
increase in the number of vibrational modes. This limitation
restricts the VCI method for the application of small molecules
and opens up the possibility of including the correlation in the
VSCF method, which was first introduced by Gerber and co-workers
in the VSCF-PT2 algorithm. To this end the core VSCF or the
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VSCF-PT2 method and its accelerated algorithm introduced by
Pele et al.67,68 which is analogous to the Møller–Plesset second
order perturbation theory (MP2) of the electronic structure
method are fairly efficient for relatively large molecules, and
were therefore, applied to a number of biological building
blocks such as amino acids, polysaccharides, proteins, etc.
One of the oldest methods applied for the treatment of
anharmonicity in molecules is the perturbation theory method.
Different perturbation theoretical methods were applied earlier
for the calculation of anharmonic levels of coupled molecular
vibrations.48,74 In the standard Rayleigh–Schrödinger perturbation
theory (RSPT), the HO approximation is considered as zero-order
Hamiltonian and the anharmonic part of the potential is treated as
coupling. The potential is expanded as a polynomial up to quartic
terms for the perturbation. This method is commonly confined to
first and second order perturbation theoretical treatment and
works better for weak anharmonic systems.75–78 Another superior
method which has several theoretical advantages over the RSPT
method is the canonical Van Vleck perturbation theory (CVPT).
It was developed by Sibert and McCoy.79,80 However, the
algorithmic structure of this method is relatively complex and
initially applied only for small systems. Later, Sibert and
co-workers successfully applied it for the OH stretch spectrum
for carboxylic acid dimers81 and CHBr3 and its deuterated
analogue82 using further approximation in the treatment. Recently,
Barone has introduced an algorithm using a perturbation
theoretical approach to treat the anharmonicity and IR-intensities83
and applied it for organic molecules such as furan, pyrrole,
uracil84–86 and other large systems.87,88 This algorithm has been
included in the GAUSSIAN suit program package.89 There are a
few other second order perturbation theory based approaches
applied successfully for ro-vibrational systems.59,75–77,90 The
important advantages of this type of theory are its simplicity and
computational efficiency in its standard form. The disadvantage of
this perturbation theoretical method is that it depends on the
accuracy of the HO approximation which is the starting point.
Since the anharmonic correction and hence the corresponding
potential is considered as the perturbation on top of the harmonic
part, for highly anharmonic systems the perturbation is expected
to be large and that may break down the method. Most of the
studies using the polynomial expansion, generally up to quartic
terms, are not adequate for floppy molecules, and in principle one
can use other representations of the potential. However, the
polynomial expansion beyond the quartic terms is computationally
very expensive and frequently shows numerical convergence
problem in the SCF procedure. Moreover, in some cases, perturbation theory only up to second order may not be sufficient and
higher than the second order is computationally very costly.
The Molecular Dynamics (MD) simulations at finite temperature
are one of the important approaches for the description
of vibrational spectroscopic properties.91,92 These methods
have the advantage of dealing with very large molecules with
considerable accuracy. Though the classical MD93–96 has been
used for long, some recent applications of ab initio MD97–101
posit a much increased signature in this field. High temperatures
in gas phase and condensed phase experimental IR spectroscopy
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make it a useful theoretical tool in order to calculate the
vibrational spectra with acceptable accuracy. Moreover, the
dynamical properties of the molecules and their effects on
the vibrational spectra can be described by these methods. In
ab initio MD the forces (and also the dipole moment) are
computed on the fly by first principles electronic structure
calculations to evaluate the vibrational spectra including
temperature and environmental effects. But this approach is
computationally costly. A useful in-depth review for this
method is reported by M. P. Gaigeot.102
Rigorous approaches to solve the vibrational Schrödinger
equations are another line of work and one example of such an
approach is the grid based methods.103–108 In principle these
methods are numerically exact. However, the computational
costs for these rigorous methods are extremely high, and as a
consequence, can only be used for small systems. For example,
recent developments by Carrington and co-workers using quadrature
grids for solving the vibrational Schrödinger equations are
noteworthy.109–112
The Diffusion Quantum Monte Carlo (DQMC)113 approach
and some other related methods have been used mainly for the
calculation of vibrational ground states. These methods are
successfully used to compute delocalized and floppy systems114–116
such as small anharmonic 4He clusters and biological molecules
like nucleobase–water complexes. This method has the superiority
of being rigorous and can be converged to achieve exact results
numerically. But the approach is not easily applicable to excited
state calculations, and additional approximations and assumptions
are required for this purpose.
Another computationally promising approach, mainly developed
by Miller and co-workers, is based on the semi-classical methods.117
They used semi-classical initial value representation118,119 to
calculate the spectral density, which showed good agreement
with quantum calculations. In general, semi-classical approximations for a multidimensional system produced reliable results.
However, the major problem with this approach is the high
demand of computational time needed to calculate the vibrational
states and it is a daunting task for systems with many degrees of
freedom.
The structure of this perspective is as follows. Section II
describes the VSCF methodology and its variants. Some new
algorithmic developments mainly used for the large biological
molecules are described in Section III. In Section IV, a few
aspects of vibrational frequencies, intensities and bandwidths
and their importance are described. Section V illustrates a few
applications and examines the validity of the new algorithmic
developments. Finally, in Section VI some open problems and
future prospects are discussed.
II. General VSCF method and its variants
The VSCF algorithm is the core method used in this approach.
The physical concept of VSCF is simple. In this approximation,
each vibrational mode is characterized by moving in the mean
field of the rest of the vibrational motions. Within this mean
field approximation, the wave functions of different modes are
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determined using a self-consistent method. This approach is
equivalent to the Hartree method used for many electron
systems. The total wave function of VSCF approximation is a
product of single mode wave functions. Hence, if the coupling
between the modes is strong then the separability approximation
may breakdown due to the correlation effect between the vibrational
modes. Historically VSCF was developed using the normal
coordinates. But it is an important issue to choose the advantageous coordinate system which can best reproduce the mutual
separability for VSCF approximation. It is found that for
low energies (near the bottom of the potential well) where
the amplitudes of vibrations are expected to be small, the
commonly chosen normal mode coordinates representation
works well. But for higher excitation energies or for extremely
anharmonic systems it usually gives poor results. For example,
the VSCF method with normal mode coordinates often fails for
soft torsional motions (such as torsional motion of –CH3)
where the couplings between the torsional modes and other
normal modes are large. The usage of other optimal coordinates was studied earlier.120–122 Horn et al.122 found that VSCF
in hyperspherical coordinates works much better than VSCF in
normal coordinates for XeH2,123 H2O,124 and CO2,125 and for
Ar3.126 Ellipsoidal coordinates were successfully applied in
VSCF for HCN 2 HNC isomerisation127 and I2He systems.128
Earlier a work by Truhlar and co-workers129 and very recently
Yagi et al.130 showed the possibilities of determining optimized
vibrational coordinates for VSCF and VCI methods. However,
one major disadvantage of the non-normal coordinates is that
these do not have any general functional form due to mathematical
complexity. Consequently, these are only applied on some
limited small systems. Some attempts have been made for VSCF
approximation using internal coordinates.131–133 Recently Suwan
and Gerber134 have shown the possibility of an alternative
approach by using curvilinear internal coordinates for the VSCF
separability, and applied it for HONO, H2S2 and H2O2 molecules
with good accuracy over the normal coordinates representation
of VSCF. Benoit and co-workers135 have studied generalized
curvilinear coordinates with a significant improvement in the
VSCF/VCI method for the torsional modes of methanol. But still
by far the most widely applicable and computationally efficient
implementation for the VSCF method is carried out in normal
mode coordinates representation due to its simplicity in the
mathematical form that defines the underlying potential of a
system. There are no VSCF applications of other coordinate
systems as yet for biological molecules. Here the general VSCF
method and its existing variants are discussed briefly.
(a) VSCF algorithm
In the first step, the Born–Oppenheimer approximation is
invoked which separates the electronic and nuclear motion.
Using an electronic structure calculation, the minimum energy
configuration is obtained. Considering a system of zero total
angular momentum ( J = 0) and neglecting the rotational
coupling (Coriolis coupling) the normal mode displacement
coordinates from that minima are determined. The Coriolis
effects are neglected because for large systems the rotational
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and vibrational motion are nearly decoupled as the Coriolis
coupling coefficients are inversely proportional to the moment
of inertia. The Schrödinger equation for the remaining vibrational
problem can be written in the mass weighted normal coordinates
Q1, Q2. . . as,
"
#
N
1X
@2
ðnÞ
þ
V
ð
Q
;
.
.
.
;
Q
Þ
Ci ðQ1 ; . . . ; QN Þ
1
N
2 i¼1 @Qi2
(1)
ðnÞ
¼ En Ci ðQ1 ; . . . ; QN Þ;
where V is the potential energy function of the system, n is the
state number and N is the number of vibrational degrees of
freedom. The VSCF approximation is based on a separable ansatz.
The N mode trial wave function is approximated as,
CðQ1 ; . . . ; QN Þ ¼
N
Y
ðnÞ
ci ðQi Þ;
(2)
i¼1
where the single mode wave functions c(n)
i are called the modals.
Here the HO approximation is avoided. However, error due to
introducing the separability approximation depends on the
coordinate system used. If a system is not very far from a
harmonic one then the normal mode coordinates provide good
approximation, at least for low lying vibrational states. Moreover
the accuracy of eqn (2) also depends on the choice of the
variables that are being factorized.
Using a variational principle for the ansatz in eqn (2) leads
to the single mode VSCF equation136–138
1 @2
ðnÞ
ðnÞ
ðnÞ ðnÞ
þ
V
ð
Q
Þ
ci ¼ ei ci ðQi Þ;
(3)
i
i
2@Qi2
ðnÞ
V i ðQ i Þ
for mode Qi is given by,
where the effective potential
*
+
Y
N
N
Y
ðnÞ
ðnÞ ðnÞ V i ðQ i Þ ¼
cj Qj V ðQ1 ; . . . ; QN Þ cj Qj : (4)
jai
jai
Here, eqn (3) and (4) must be solved self-consistently for the
single mode wave functions, energies and effective potentials.
Several methods can be applied for the solution of eqn (3) to get
both the ground and excited VSCF states of the system. Due to
this approximation the total energy is given by,
*
+
Y
N
N
N
X
Y
ðnÞ
ðnÞ ðnÞ ei þ ðn 1Þ
c j Q j V ð Q 1 ; . . . ; Q N Þ c j Q j :
En ¼
jai
jai
i¼1
(5)
The major computational difficulty is due to the evaluation of
multidimensional integral inherent in eqn (3)–(5), especially for
large systems, and that depends on the mathematical form of
the potential. Hence the choice of potential plays a key role in
the VSCF approximation.
(b)
Representation of the potential
Selection of the functional form of the underlying potential is
an important issue for the applicability of a method. The first
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Phys. Chem. Chem. Phys., 2013, 15, 9468--9492
approach in this direction is to use a power series expansion of
the potential in normal mode coordinates,
X
V ðQ 1 ; . . . ; Q N Þ ¼
Vm1 ;...;mn ðQ1 Þm1 ðQN Þmn :
(6)
m1 ;...;mn
That leads to an evaluation of one dimensional integral in order
to obtain the single-mode effective potentials. Due to its
simplification and if the higher terms are not included in the
potential, the numerical efforts in evaluating the integrals are
not large. A fourth order polynomial or quartic force-field is the
standard approximation to represent the potential.136 However,
for practical purposes semi-diagonal quartic potential (i.e. Viijj)
is used for several test cases to reduce the computational time.
Using this the evaluation of single mode effective potential is
an important advancement and hence used for several VSCF and
CI-VSCF applications.139–141 However, for a strong anharmonic
system with several floppy modes such as a hydrogen bonded or
a van der Waals cluster, the power series expansion in normal
modes either diverges or converges very slowly. Additionally, as
the potentials are not explicitly available in analytical form,
calculation of higher order terms is extremely costly.
To deal with such difficulty, an alternative representation of
the potential has been introduced by Jung and Gerber142 that
has been applied successfully in VSCF and VSCF-PT2 approximations. In this approximation the potential is written as a
sum of terms that include single-mode potentials and pair-wise
interaction between normal modes:
V ðQ1 ; . . . ; QN Þ ¼
N
X
Vidiag ðQi Þ þ
XX
j
i¼1
Wijcoup Qi ; Qj : (7)
i4j
Here, the potential at equilibrium is conveniently taken as zero.
Vdiag
(Qi) are the single-mode diagonal terms and defined by,
i
(Qi) = V (0,. . .Qi,. . .,0),
V diag
i
(8)
and the pair-wise interactions are
(Qi, Qj) = V (0,. . .Qi,. . .,Qj,. . .,0) – V diag
(Qi) – V diag
(Qj).
W coup
ij
i
j
(9)
(Qi) and W coup
(Qi, Qj) are usually obtained by computing
V diag
i
ij
the potential function along Qi keeping the other mode at
equilibrium and potential for different Qi, Qj keeping all other
modes l a i,j at equilibrium. Due to this two dimensional
quadrature in the coupling terms, the scaling of the coupling
effect becomes proportional to N2 but this is still affordable
even for a large system. As a further approximation to calculate
the pair wise potential faster, Pele et al.68 proposed to choose
the number of important coupling terms prior to evaluating all
the terms. This can reduce the computational time extensively
with good accuracy. Till now, the agreement of the computed
spectrum with experiments supports the pair wise interaction
nicely. Note that, it is also possible to extend this representation by adding a limited number of triplets or quartets of
normal modes for higher accuracy. Inclusion of a small set of
higher terms will not affect the computational terms to a large
extent, at least for small or moderate sized molecules with a few
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degrees of freedom. However, full inclusion of all terms (up to
quartic) is not desirable. Inclusion of the coupling of triplets of
normal modes was tested for H2O and Cl (H2O).143 It was
found that the contribution is non-negligible, but not too
significant to justify the increased cost of CPU time. While this
approximation must be tested on a case by case basis, it has been
applied extensively for many VSCF calculations of biological
molecules56,144–151 with satisfactory results. The VSCF algorithm
was introduced in the GAMESS152 suit of program, and later also
in the MOLPRO153 program package.
(c)
VSCF-PT2 method
Gerber and co-workers proposed an approach where the results
of the VSCF approximation can be further improved with a
second order perturbation correction on top of it.65,142 The idea
behind this approach is that the difference between the true
Hamiltonian and the VSCF Hamiltonian must be small as VSCF
is found to be good approximation. So, it is acceptable to
consider the difference as a perturbation. The full Hamiltonian
is written in the form
H = HSCF,(n) + DV (Q1,. . .,QN,),
(10)
where HSCF,(n) is the VSCF Hamiltonian written as,
N X
1 @2
ðnÞ
þ
V
ð
Q
Þ
:
H SCF;ðnÞ ¼
i
i
2 @Qi2
i¼0
(11)
The equation for DV is given by,
DV ðQ1 ; . . . ; QN Þ ¼ V ðQ1 ; . . . ; QN Þ N
X
ðnÞ
Vi ðQi Þ:
(12)
i¼1
The difference between the correct Hamiltonian and the VSCF
Hamiltonian and hence the correlation effects are all included
in DV. Assuming this term as sufficiently small, the second
order perturbation theory can be applied as,
EnPT2 ¼ En0
þ
N
X
DQ
ðnÞ
ei þ
i¼1
Q
ðnÞ ðnÞ N
Qj V ðQ1 ; . . . ; QN Þ N
Qj
jai cj
jai cj
ð0Þ
ð0Þ
En Em
E
;
(13)
where EPT2
is the correlation corrected energy of state n and E0n
n
is the VSCF energy. It has been found that this second order
correction contributes significantly in many cases to improve
the agreement with experiment.57,143
(d)
Direct ab initio VSCF versus fitting potentials
Other than the direct ab initio VSCF method mentioned above
for vibrational spectroscopic calculations, there is another
interesting line of approach available in the literature where
spectroscopic calculations for an ab initio potential is carried
out by fitting the potential to a suitable analytical form. In
principle any vibrational spectroscopy method can be used for
this approach. Such methods have been investigated in the past
and still continue to be a very active and successful route in
this field. However, due to some unavoidable problems, this
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method is restricted to less applicability. It was found that high
quality fitting is necessary for accurate spectroscopic data in
this approach. This fitting procedure is a formidable task and
computational demand increases almost exponentially with the
increase in the dimensions of the system. That restricts the
fitted ab initio potentials which were generated only for a few
atomic molecules such as cis- and trans-HONO.154,155 In those
cases rigorous methods were used with good quality fitting for
the potential results in very good agreement for the frequencies with
the experimental values. Moreover, it is difficult to find a generally
accepted algorithm for the construction of fitting potential. But this
rich field of fitting methodology is still very much active with many
strategic developments and applications.156–160 Bowman and
co-workers presented a few nice examples of CH5+,157 hydrated
chlorides161 and some others162–164 using fitted ab initio
potentials. Presently, it seems that this direct fitting method
can be used efficiently for 9–10 atomic molecules, not for
biological molecules such as peptides and sugars.
(e)
Treating degeneracy
The VSCF-PT2 method is based on the non-degenerate perturbation
theory. Occasionally it can suffer from unphysical large correction of
energy due to the near degeneracy. For degenerate vibrational mode
it caused singularities due to the zero denominators in eqn (3).
Matsunaga et al.165 introduced two variants of VSCF for computing
vibrational transition when 1 : 1 resonance occurred. For a simple
algorithm, VSCF-VCI performs diagonalization of the vibrational
Hamiltonian on a VSCF basis over the degenerate subspace. In a
second algorithm VSCF-DPT2 beyond diagonalization over the 1 : 1
resonance space is performed considering interaction with nondiagonal states using second order perturbation theory. Another
simple treatment of VSCF-PT2 variant to deal with degeneracy is to
exclude all terms with a denominator smaller than a critical value. It
is implemented in GAMESS to avoid the problem of singularities.
Daněček and Bouř166 introduced another treatment by replacing the
denominators with square roots according to the Taylor series. This
method is referred to as VSCF degeneracy corrected perturbation
theory of the second order (VSCF-DCPT2). There are some other
interesting studies in this context by Yagi et al.167 and Respondek
and Benoit168 which should be noted to handle the quasi degeneracy, and recently by Barone169 where the DCPT2 method is further
improved to a hybrid version (HDCPT2) for more reliable results.
(f)
Post VSCF methods
Considerable progress has been observed in the development
of post VSCF calculations and applied mostly for small molecules.
The multi-configurational treatment of VSCF (VMCSCF) is one
important line of approach in this direction. Initially it was
developed by Culot and Lievin170 in a time independent context.
Recently Heislbetz and Rauhut171,172 implemented it successfully
and showed its accuracy. In a time dependent context, a multiconfigurational time dependent hartree (MCTDH) approach173
draws great attention over the years to calculate the vibrational
levels with high accuracy.
Among all the methods available in the literature, the
vibrational configurational interaction (VCI) method71,73,174
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gives variationally the best possible results within the basis set
limits. Primarily it was developed to count the correlation
among the vibrational modes in the VSCF approximation to
improve its result. In VCI methodology, every possible contribution of a complete set of functions is considered and, as a
consequence, the full VCI with an infinite basis set is the exact
solution of the time independent Schrödinger equation.
Although it gives very accurate results, it suffers from high
computational costs. Some recent further developments in VCI
methodology175–182 have been noticed. In an interesting paper,
Neff and Rauhut180 proposed state specific configuration
selected VCI for fast evaluation of state energies and showed
the possibilities to calculate as large as 15 atomic molecule.
Vibrational Møller Plesset (VMP) perturbation theory up
to second order was developed by Gerber and co-workers
(the VSCF-PT2 method mentioned earlier). A general order
VMP theory was developed by Christiansen90 along with lower
computational scaling.183,184 However, higher than second
order VMP has not been used so far, due to the computational
effort required.
One of the most important methods explored as post VSCF
calculation is the vibrational coupled cluster method (VCCM).
Initially, it was proposed for one dimensional case using harmonic
oscillator reference states.185,186 For a coupled anharmonic oscillator
VCCM was developed by Prasad and co-workers187 using bosonic
representation.188 Later, Christiansen189,190 developed it for basis set
representation and successfully applied for small molecules.190
Recently vibrational multi-reference coupled cluster theory
(VMRCCM) has also been developed by Prasad and co-workers.191
However, all the above mentioned methods are intensely computational time demanding processes and are only applied for small
molecules. Though the architectural power of computers increases
rapidly, still it seems to be a difficult task to use a post VSCF method
for biological molecules even in near future.
Some related developments in VSCF theory by introducing temperature into the anharmonic treatments of vibrations and especially
calculation of partition functions are noteworthy.131,192–194 A few
methods have been developed to calculate the thermal averages of
molecular properties using VSCF methodology. One is the state
specific VSCF (ss-VSCF) method developed by Christiansen and co
workers192 where the single mode potential is optimized and then
VSCF calculation is performed for each state. As a further approximation, a virtual VSCF (v-VSCF) method was developed where the
VSCF is performed only for the ground vibrational states and the
virtual modal energy differences in this vibrational ground state are
considered as excited state energies to calculate the partition function. Recently, Roy and Prasad193,195 proposed a few alternative
approaches where the thermal density matrices have been used as
a separable ansatz to calculate the partition functions using the
Feynman variational principle based on Gibbs–Bogoluibov inequality193 and the McLachlan type variational principle.195
(g)
Applicability of VSCF and comparison with other methods
In recent years all these methods mentioned above have been
used extensively for anharmonic calculation of the vibrational
spectra of several systems with different size. But for biological
9474
Phys. Chem. Chem. Phys., 2013, 15, 9468--9492
molecules, to date, VSCF and its variants are the most widely
used tools. This is due to the fact that all the post VSCF
methods are computationally much more expensive and hence
are only applied for small systems, and are mostly a hope for
the future for biological molecules. Additionally the algorithmic
structures of some of the post VSCF methods are complicated to
implement. On the other hand, VSCF is fast, easy to implement
and gives good agreement with experiment even for macromolecules, in particular biological molecules. The force-field
method is another feasible option for biological molecules. It
is indeed true that this method is much faster than the VSCF and
can be applied for very large molecules. But it is found that56
spectroscopic accuracy of standard force-fields such as AMBER,
OPLS-AA and CHARMM is not good enough for biological
molecules and performed poorly to define soft vibrational
modes, hydrogen bonding, etc. Chaban et al.143 showed that
the spectroscopic accuracy of the ab initio potential is much
superior than the state-of-the art empirical potential OPLS-AA for
the three conformers of glycine. Table 1 shows the spectral data
for three lowest energy glycine conformers using VSCF-PT2,
OPLS-AA and experimental values. It is found that OPLS-AA
produced very similar vibrational frequencies for all the three
conformers. That reflects the fact that this method is not capable
of distinguishing among the potential underlying each conformer, which is in disagreement with the ab initio or experimental
results. The main reason behind this observation is that this
empirical force-field (and others like CHARMM and AMBER) is
not able to describe the intramolecular hydrogen bonding
present in glycine due to lack of parameterization for such
interactions and hence is not useful for biological systems. This
result directs us to the fact that the ab initio force field is needed
Table 1 Vibrational frequencies (cm1) for glycine conformers: VSCT-PT2, OPLSAA and experiment56
Conformer 1
Conformer 2
Conformer 3
Mode Ab initio OPLS Expt Ab initio OPLS Expt Ab initio OPLS Expt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
3598
3382
3343
2986
2959
1805
1669
1473
1410
1377
1290
1185
1167
1122
970
943
847
633
613
514
463
352
270
143
3701
3279
3259
2953
2899
1628
1592
1494
1381
1246
1093
1070
1044
991
968
838
753
571
455
449
390
329
318
176
3560 3270
3410 3428
3360
2989
2958 2958
1779 1824
1630 1653
1429 1483
1373 1399
1363
1317
1219
1136 1166
1101 1073
907 976
883 926
801 850
619 849
648
500 565
463 508
329
323
144
3662
3277
3256
2947
2887
1638
1607
1448
1401
1283
1111
1079
1030
1007
892
872
781
769
556
472
355
349
268
169
This journal is
3200 3612
3410 3393
3360
2955
2958 2931
1790 1800
1622 1754
1429 1595
1390 1413
1362
1346
1210 1207
1130 1163
1144
911 1032
880 938
786 817
734
638
593
463 499
280
241
67
c
3683
3279
3257
2952
2895
1636
1586
1445
1393
1269
1080
1045
1040
994
947
855
754
550
427
442
387
325
321
145
3560
3410
2958
1767
1630
1429
1339
1147
1101
883
777
463
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for acceptable accuracy. In recent days the DFT based MD has
been found to be an interesting alternative for the spectroscopic
calculation of biological molecules. It has the superiority to
handle the large systems with the inclusion of temperature,
the phase of a system and other environmental effects. A low
level ab initio method such as DFT/BLYP is a common choice for
this purpose for faster calculation. But it is often found that this
kind of functional is not accurate enough and going beyond a
higher level such as B3LYP it is too costly for DFT-MD, except for
small biological molecules such as monosaccharides.196 Moreover, the initial conformational samplings of the potential
energy surface, aqueous phase DFT-MD for long peptide chains,
inclusion of vibrational anharmonicity in a direct way are some
existing problems which often produce unsatisfactory results.
Thus, currently VSCF (and its variants) is the stand alone
algorithm which can keep a balance between spectroscopic
accuracy and CPU time to a good extent for biological molecules.
III. Algorithmic developments
All the post VSCF methods mentioned above suffer from high
computational costs and hence applied to small molecules with a
few vibrational modes. VSCF and VSCF-PT2 are such methods that
are computationally cheap and can be applied for medium sized
molecules (B30 atoms) with good accuracy. However, calculations of
very large molecules (>50 atoms) such as proteins, peptides and
sugars need high computational time even for core VSCF calculations. The main bottleneck of direct VSCF calculation is the
construction of the potential energy surface (PES). An ab initio
computation is needed for each grid point of the PES. The grid point
increases rapidly with the increase in the number of vibrational
modes. Consequently, the main time consuming part is the calculation of the pair-wise interactions or coupling potentials, W coup
(Qi, Qj).
ij
For VSCF-PT2 the computation time is even more, since eqn (13)
requires the additional calculations of a large number of integrals for
several vibrational excited states. Thus, to handle such situations, a
few algorithmic developments have been carried out to perform fast
VSCF or VSCF-PT2 calculations for large biological molecules. In this
section some of such developments are described that have improved
the computational time with acceptable accuracy.
(a) N3 acceleration
Substituting eqn (14) in the numerator of eqn (13) leads to the
cancellation of diagonal terms. That yields
*
+
N
N
Y
Y
ðnÞ ðmÞ cj Qj DV cj Qj
j¼1
j¼1
¼
XZ
ðnÞ
ðnÞ ci ðQi Þcj
ðmÞ
ðmÞ Qj Wijcoup Qi ; Qj ci ðQi Þcj Qj dQi dQj ;
j4i
(15)
where, the integration is over the normal modes i and j, and m
is the label of the excited vibrational states (m a n). Here
the orthonormality of the single mode wavefunction is also
considered to reduce the equation in its current form. Note
(Qi) and c(n)
(Qj) are not strictly, but very nearly,
that, c(n)
i
j
orthogonal. However, it was tested and orthogonality holds in
this case with very good accuracy. As a consequence of this
algorithm, the computational time of the second order correction terms in the VSCF-PT2 calculation goes down dramatically
from 85–95% to 0.5–2.5% of the total run time. The computational time for the second order correction terms, which was
originally roughly O(N4), is reduced to O(N2 + c * N4) where c is
very small constant close to zero. Thus the overall runtime is
reduced from O(N6) to O(N3). Fig. 1 shows the runtime for the
VSCF/VSCF-PT2 calculations of glycine (10 atoms), diglycine
(17 atoms), triglycine (24 atoms) and tetraglycine (31 atoms)
molecules. It indicates that the runtime improvement is greater
for larger molecules. For glycine the improvement is a factor
of 5.9 and for tetraglycine it is a factor of 16.5. Thus this
elegant implementation of acceleration made it possible to
apply VSCF-PT2 for large biological molecules.
(b)
Number of coupling
In another study, Pele and Gerber68 developed an algorithm to
reduce the number of coupling terms in the VCSF-PT2 to accelerate
the calculation significantly. The improvement in the computational
time for the second order perturbation theoretical correction terms
as stated above leaves eqn (7) as the main bottleneck for the VSCF
calculation. This scales N2P where P is proportional to the CPU time
for a single potential grid point. Hence the major time consuming
part is the calculation of coupling terms. Note that P, mentioned
above, depends on the method used for the calculation of the grid
point. For example, DFT variants scale as P = M3 to M4 where M is
Pele et al.67 developed an important acceleration in the VSCFPT2 method. In this algorithmic development, the orthogonality of the single mode vibrational wavefunction was employed
to reduce the number of integrals needed to calculate the
second order perturbation theoretic correction terms of
eqn (13). Considering the pair-wise approximation terms in
eqn (12), it can be re-written as,
DV ðQ1 ; . . . ; QN Þ ¼
N
X
Vidiag ðQi Þ þ
XX
j
i¼1
Wijcoup Qi ; Qj
i4j
(14)
N
X
ðnÞ
Vi ðQi Þ:
i¼1
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Fig. 1 Running times for VSCF/VSCF-PT2 in GAMESS as a function of number of
atoms in glycine peptides. Data from ref. 67. Reprinted with permission from
ref. 67. Copyright 2008, Springer.
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the number of basis function used. For HF calculation P = M4 and
for CI and MP calculations P = M5 or higher. This algorithm is based
on the assumption that most of the normal-mode pairs have a very
small coupling potential. Only some groups of normal-mode pairs
have relatively strong coupling potential and hence contribute to the
major part in the results. Earlier Leitner197 pointed out the same
observation for the five largest a helices of myoglobin. Thus it was
assumed that modes involving a similar magnitude of displacement
of the same atom have large mutual coupling. Using the statistical
tool, Spearman’s rank correlation coefficients, the predicted important couplings (PICs) are assigned for the calculation of the
potentials. It was found that PIC improves the acceleration of
VSCF-PT2 significantly with good accuracy by reducing the number
of mode–mode coupling terms from N2 to N log N. Fig. 2 shows
absolute coupling potential of all normal mode pairs of glycine,
tetraglycine and ValGlyVal. It indicates that most of the normal
mode pairs have very low coupling potential and only some of them
show the high coupling potential. Using this algorithm they have
shown that a large biological molecule like ValGlyVal with 120
normal modes produced results with considerable accuracy and
significant speed up.
Though the N3 acceleration and the PIC method decrease the
computational time for VSCF-PT2 against VSCF to a great extent, it is
always more time demanding than the VSCF. This additional
computational time increases with the increase in the number of
vibrational degrees of freedom. However, Pele and Gerber198 have
shown that the mean deviation of VSCF frequencies from VSCF-PT2
frequencies decreases with the increase in the number of vibrational
modes. This conjecture is a manifestation of improved mean
accuracy of VSCF as a mean field approximation, since more degrees
of freedom lead to more extensive averaging. They have shown this
trend for a series of amino acids and peptides. A systematic increase
in accuracy of VSCF with an increase in the number of vibrational
modes is found for certain groups of transitions, such as N–H
stretching which are important modes for biological molecules.
Fig. 3 shows the mean deviation between the frequencies of VSCFPT2 and VSCF as a function of number of vibrations. It shows a clear
trend of decreasing deviation, which encourages to use VSCF for
large proteins, peptides, polysaccharides and nucleic acids.
(c)
Other developments
An interesting improvement in the acceleration of the VSCFPT2 algorithm was proposed by Benoit.199 In general, it is
observed that weakly coupled vibrational modes produce small
changes in the potential energy of the system and can be
neglected to a good approximation. In this method, the negligibly small pair-wise coupling between the normal modes is
estimated. Then those weakly coupled terms are omitted from
the calculation and only the important terms are considered. As
a consequence of that a large reduction of effort compared to
VSCF-PT2 was achieved. The relative magnitude of the coupling
between modes can be calculated as
x Qi ; Qj ¼
nmax X
nmax
X
jWijcoup ni ; nj j;
ni ¼1 nj ¼1
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Fig. 2 Histogram of the absolute coupling potential of all the normal-mode
pairs for glycine, tetra glycine and ValGlyVal. Reprinted with permission from
ref. 68. Copyright 2008, American Institute of Physics.
where n is a set of points distributed along modes Qi and Qj
forming nmax*nmax grid points.
If this value is less than some pre-assigned threshold values
then coupling terms are considered as zero, and ab initio
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where VLM is the potential at the low-level method (LM) and
VALM is the potential at the adjusted low-level method (ALM).
Here the scaling coefficients li are defined as the ratio of the
harmonic frequencies of the low-level method (LM) to those of
the high-level method (HM),
li ¼
Fig. 3 The mean deviation between the VSCF-PT2 and VSCF frequencies as a
function of number of modes. Averaging over all the fundamentals. Reprinted
with permission from ref. 198. Copyright 2008, American Chemical Society.
calculation for that grid point is skipped. That leads to a
considerable speed up and could be very useful for the calculation
of large molecules. In another study, Benoit200 proposed to select
the active vibrational modes of interest and calculate the pair-wise
coupling terms of those vibrational modes with all other modes.
The rest of the inactive vibrational modes are treated as noncoupled anharmonic oscillators. This assumption was based on
the observation that most studies are focused on a small part of
the vibrational spectra than the total spectra, such as an OH
starching frequency region in amide bonds or hydrogen bonds of
a protein. This approximation leads to a much faster algorithm
than the standard VSCF calculation and can be applied for large
molecules of particular interest. Some other interesting approximations for the runtime improvements were proposed by
Benoit,201 Rauhut53 and Yagi et al.184
(d)
wHM
i
:
wLM
i
(18)
Thus the scaling coefficients are chosen so that the improved or
adjusted potential reproduces the frequencies of the high level
method for the potential at the low level method. For example,
if the low-level method is semi-empirical but computationally fast
algorithm like PM3 and the high-level method is ab initio but
computationally slower like MP2, then improved VALM can be
derived almost at the cost of the PM3 level of calculation. This
method was successful for many spectroscopic calculations144,148–150
of large molecules mostly using PM3 or HF as the lower level and
MP2 as the higher level. To be physically reasonable, the scaling
should be applied when the normal modes of the low-level and
high-level potentials are similar. Overlap of the two normal modes is
used as a criterion for scaling.
(e)
XVSCF method
Recently, Keceli and Hirata202 have introduced a new variant of
VSCF, stated as the size-extensive VSCF (XVSCF) method. In this
work, considering the polynomial form of the potential up to a
quartic force-field, some non-physical size dependence terms
have been eliminated for the size-extensivity consideration. As
the linear, cubic and other higher odd-order force constants
cannot contribute to XVSCF, those terms are eliminated. Since in
XVSCF formalism only a small subset of the quartic force-field was
included, the mean field potential turned out to be always
effectively harmonic in nature. Thus the equations can be solved
without matrix diagonalization. That makes it faster than the
conventional VSCF method with considerable accuracy, especially
for large molecules.
Hybrid potential in VSCF
One of the important parameters to determine the accuracy of
the VSCF and the VSCF-PT2 method is the level of electronic
structure theory used to calculate the grid points. Obviously
higher level electronic structure methods (MP2, B3LYP, CCSD
etc.) should produce better results using more computational
time and lower level methods (AM1, PM3, HF etc.) should show
less accuracy using less computational time. So some balance
between the computational time and accuracy is needed especially
for large molecules. It is not surprising to get poor spectroscopic
results produced using a semi-empirical PM3 method or a non
hybrid DFT functional BLYP method in comparison against MP2
or B3LYP for VSCF-PT2 calculations. However, PM3 and BLYP
methods are much faster than the MP2 and B3LYP. Brauer et al.144
successfully upgraded the potential surface of a relatively low level
method to produce a better anharmonic surface. This improved
potential is written as
VALM (Q1,. . .,QN) = VLM (l1Q1,. . .,l1QN),
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IV. Frequencies, intensities and bandwidths
The most important peak position for biological molecules is in
the domain of 3000–4000 cm1 for N–H and O–H stretching
modes and 1000–2000 cm1 for the vibrations of amide I, II and
III modes. Most of the experimental data are available for these
regions which indeed carry information relevant to the structures of the biological molecules. For example, the red shifts of
the N–H, O–H and CQO stretch modes and corresponding blue
shifts of N–H and O–H in plane bending of peptides indicate
the presence of hydrogen bonded structures. The extent of
these shifting depends on the strength of the hydrogen bonds.
The peak position of amide I (CQO stretching), amides II and
III (N–H bending coupled with C–N stretching) provides the
information on hydrogen bonding, dipole–dipole interactions and
peptide backbone geometry of secondary structural changes. Such
frequencies are in principle more characteristic of the details of
the structural information of biological molecules and can provide
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an important test for computational methodology including
anharmonicity in the potential. Thus a well defined peak
position and the nature of a peak in this domain, which is
comparable to experiments, are the signature of a better
theoretical approximation.
(a) Isolated molecules
The theoretical treatment to calculate vibrational spectra is
generally carried out for isolated molecules preferably in the
gas phase and considering 0 K. It is expected that for the
experimental techniques such as molecular beams or molecules in noble gas matrices, the isolated molecule approximation is the reasonable one since the interaction with the
environment is expected to be weak. Spectral data obtained
from these techniques are good reference for benchmarking the
theoretical approaches. But experimentally perfect isolation of
molecules is a formidable task. It needs ultracold temperature
and infinitely diluted concentration or perfect trapping in other
inert substrates, which is difficult to achieve. High resolution in
the vibrational spectra of biological molecules depends on these
experimental conditions. This is due to the fact that the potential
energy surfaces of a simple biological molecule have multiple level
minima corresponding to multiple conformers separated by low
barriers. Many of these conformers are significantly populated
even at room temperature. Moreover, the excited states of the soft
vibrational modes are also expected to be populated near the
room temperature. Additionally, environmental effects, such as
solvent or interactions with other molecules, may influence the
structure and properties of biological molecules. These effects,
when present, lead to a spectrum with poor resolution and broad
peaks. However, low temperature reduces the possibilities for
high energy conformers to be populated and affects the spectrum.
Specifically, at ultracold temperature it is possible to get the
spectrum for the lowest energy conformer or a small number of
conformers. Very low concentration of the molecule reduces the
effects of the interaction with other molecules. Under these
conditions, it can be considered as an isolated molecule. It gives
best possible high resolution data to compare with the theoretical
treatment. Most of the modern experimental techniques
mentioned earlier generally secure these conditions for better
spectroscopic results. Hence high resolution spectra obtained
by these techniques need accurate computational theory to
interpret them. But, these state-of-the-art computations face
challenges to calculate accurate spectra with the increase in the
number of atoms. Computation of isolated molecules without
solvent and other environmental effects makes high-level treatment of the vibrational problem more feasible.
In a common quantum mechanical calculation like VSCF
and others, the geometry optimization using an ab initio
method and subsequent vibrational analysis are performed
considering a single molecule, in the gas phase at 0 K. The
generated theoretical spectrum is then compared with the
experimental data. This approach is also used for conformational search in terms of position and relative intensities of the
spectral bands. That helps to identify which isomer(s) can be
responsible for the spectra obtained using an experimental
method. Fig. 4 shows the comparison of the jet cold molecular
beam experimental data and VSCF-PT2 spectral data for the
O–H stretching frequency domain of three different conformers
of phenyl-beta-glucose.203 It is seen that the theoretical spectra
match with good accuracy the experimental spectra. Note that
the experimental spectra always suffer from broadening effects
and for biological molecules this effect is more. The spectral
width is relevant mostly to biological molecules due to the
presence of several conformers, hot bands, vibrational energy
redistribution, etc. One can minimize the broadening but can’t
remove it. However, theoretically evaluated spectra are sharp
lines. Thus comparison between theoretical and experimental
Fig. 4 OH vibrational ion dip spectra (left) for three lowest energy conformers of phenyl-b-glucose together with their structures using the B3LYP/6-31+g* method
and corresponding mid-IR spectra (right) from experimental (black) and theoretical (red) data. Reprinted with permission from ref. 203. Copyright 2011, American
Chemical Society.
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spectra needs some extra attention. For example, experimental
spectra measured at low temperature give sharper lines than
the spectra at higher temperatures. On the other hand, consideration of some width parameters in the calculated spectra
makes it broader for the comparison with experimental data.
(b)
IR, combination modes, overtones, Raman, etc.
As stated above, the experimental absorption or emission
spectral lines are not infinitely narrow. There must be some
broadening inherent in the line shapes. There are several
reasons for this broadening. Natural broadening is one of
them, where it is referred to as natural line width. It is intrinsic
to the transition and resulting from the Heisenberg uncertainly
principle where the finite life time (t) is associated with an
uncertainly in the energy of the excited states. It is very much
possible that one vibrational energy level is spread out and
hence transition between any two energy levels does not
correspond to an exact energy difference. Consequently, absorption or emission does not correspond to an exact frequency but
over a range. Depending on the nature of the vibrational mode that
range can be broad or sharp. Temperature (Doppler broadening)
and pressure (collision broadening) are also the other two factors
for the spectral width. Theoretical treatments produce the position
of a peak with intensities, which generate a spectrum of sharp lines
that are smoothed by convolution with a Lorentzian or Gaussian
of some reasonable width (B10 cm1). This model introduces
the broadening effects in theoretical spectra present in the
experimental systems.
During the computation of potential energy, the anharmonic
IR intensities are evaluated using calculated dipole moments.
For fundamental and overtone excitations the intensity is
expressed as,
E 2
8p3 NA D ð0Þ
ðmÞ
Ii ¼
oi ci ðQi Þ~
uðQi Þci ðQi Þ :
(19)
3hc
Here u is the dipole moment’s vector, oi is the vibrational
frequencies that can be calculated by VSCF or VSCF-PT2 for the
(m)
normal mode i. c(0)
are the VSCF wave function for the
i and ci
ground and the mth excited vibrational states, respectively. The
expression for combination excitations of mode i and j is
given as,
E2
8p3 NA D ð0Þ
ðmÞ
ð0Þ ðmÞ Ii ¼
oij ci ðQi Þcj Qj ~
uðQi ; Qj Þci ðQi Þcj Qj :
3hc
(20)
Here, m and n are excitation levels for modes i and j. The
backscattering non-resonance Raman intensities are calculated
using an harmonically derived approach. Only the frequencies
are the anharmonic parameters. Here again the theoretical
treatment such as VSCF calculation yields sharp transition
frequencies. Thus all effects of homogenous and inhomogeneous broadening are treated by considering each transition
to a corresponding Lorentzian or a Gaussian band. This produces
a smooth spectrum for a reasonable width. Fig. 5 shows the IR
spectra including the combination modes and Raman spectra of
Butane.204 The agreement of calculated IR and Raman spectra
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Fig. 5 IR and Raman spectra of the energetically preferred conformer of Butane
with MP2 potential (green) compared to experiments (black). The IR spectra
include also the combination modes. Data taken from ref. 204.
with experiments is very good. Here the theoretical width is
considered to be 10 cm1 using Lorentzian functions for
each case.
(c)
Temperature and environmental effects
In one of the above sub-sections we have discussed spectra of
isolated species where a single conformer of a species is
considered at 0 K temperature without any environmental
effect. It is indeed a theoretical approximation when the results
are compared with some experimental tools such as matrix
isolation spectroscopy. At very low temperature a single conformer
of a system or a few numbers of conformers can be isolated to
obtain a well resolved spectrum. Thus it is reasonable to consider
that the molecule is frozen at its lowest energy conformer and the
temperature is not sufficient enough to overcome the energy
barrier to reach other conformer(s). But at higher temperatures
this assumption is not valid and then temperature plays an
important role in the conformational dynamics, in particular
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for floppy molecules. For example, room temperature provides
enough energy to overcome the low barriers for different
conformers of a floppy biological molecule. That makes the
experimental spectra sufficiently broad, and complex to interpret.
In this situation, for theoretical treatments, one needs to introduce
temperature explicitly considering, say, a Boltzmann population
factor with a relative weight of the different conformers. This is a
standard process for the calculation of vibrational spectra using
MD simulations at finite temperature. Gaigeot and co-workers205
showed a nice example for small protonated poly-alanines (Ala2H+
and Ala3H+) by calculating vibrational spectra using finite
temperature MD simulations. They found that the DFT based
MD performed well with good agreement with experimental
values. Other than the temperature, the environmental effect is
another important issue which affects the experimental spectra
to a considerable extent. It is obvious that in life processes,
biological molecules work at ambient temperature surrounded
by various other molecules, such as, very commonly water. The
experimental tool, like noble gas matrices and the matrix
environment, does affect the spectra of the target molecule.
Thus temperature and environmental effects play a pivotal role
in the measurement of experimental spectra of biological
molecules and in most of the cases, these hinder to reveal the
intrinsic features of structures and other properties. Both the
temperature and environment are very much responsible for
the broadening of the line width and shifting of the peak
position, and multiple peaks come very close to each other due
to the presence of multiple conformers and other molecules.
Hence the theoretical treatments considering such effects are
much tricky and a well resolved spectrum is generally required
for comparison. There are a few such calculations available
where a strong interactive host medium is considered for the
target molecule to assess the environmental effects. In one
interesting work by Adesokan et al.,149 anharmonic calculations
for the Raman spectra of intermediates in the photo-cycle of
photoactive yellow protein (PYP) was examined using the VSCF
method with hybrid potential. In this study, experimentally the
system was embedded by a Raman active atmosphere and at
ambient temperature the Raman spectra of three intermediates
were explored, say, the initial ‘‘dark’’ states and two short lived
intermediates, one blue shifted and another red shifted with respect
to the ‘‘dark’’ state. In theoretical treatment, the chromophore
molecule was considered explicitly surrounded by model small
compounds that mimic the original interactions in the active site
residue. Due to this model it was computationally feasible with the
use of PM3/B3LYP hybrid potential for the calculation of spectra.
This potential showed a remarkable agreement with experimental
spectra for all three states. For example, the calculated average error
in frequencies for the red shifted intermediate was only 0.82% from
the experiment. In Fig. 6, the deviations of VSCF calculated Raman
frequencies from experiment for the M intermediate are shown.
Except one, all other deviations are quite small. These encouraging
results showed the possibility for the study of quantitative spectroscopic calculation of a biological molecule in the presence of a
protein host. It is worth noting, though, that intensities were not
computed for this case.
V. Applications
(a) Which potentials are best
For the direct calculation of anharmonic vibrational spectroscopy on a potential energy surface, the choice of potential is a
very important issue. The algorithm generates potential energy
surface points over a grid in coordinate space using a suitable
electronic structure method and subsequently uses these
points for the VSCF (or VSCF-PT2) calculation. However, with
Fig. 6 Comparison of percentage deviation of VSCF frequencies PYPM from Raman frequencies. Reprinted with permission from ref. 149. Copyright 2007, American
Chemical Society.
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the increase in the number of atoms, the number of grid points
increases rapidly. Hence it is essential that the electronic
structure method should be sufficiently fast to allow the
calculation of a large number of potential energy points. For
example, 10 atomic molecule glycine needs to evaluate around
50 000 potential points to assess the relevant range of configurations appropriate to its spectroscopy. It is essential for a
chosen electronic structure method to carry out the calculations for these many potential points using feasible CPU time.
It is found that for anharmonic vibrational spectroscopic
calculations, MP2 and variety of DFT based electronic structure
methods with a moderate level of basis work well. A semiempirical based PM3 method and some empirical potentials
are also tested for some systems with not much success. Thus
spectroscopy served to assess the validity and accuracy of
different force fields or potentials for biological molecules.
MM, QM/MM and PM3 are feasible for large biological molecules such as proteins and polypeptides since these methods have
a very fast algorithmic structure. However, the accuracy of the
calculated spectra may not be satisfactory in comparison to
ab initio methods. It should be kept in mind that accuracy and
computational cost for anharmonic vibrational calculations
depend on various parameters such as molecular size, the level
of the method, the anharmonic nature of the molecule, etc. The
subsequent discussions focus on several specific examples of such
kind. In Table 1, the comparison of glycine calculated in different
potentials is shown. Here the ab initio potential is based on the
MP2/DZP level for VSCF-PT2 calculations. Using an OPLS-AA
potential energy surface, the VSCF-PT2 frequencies are calculated
and the experimental values are taken from matrix experiments.206
It can be seen that the ab initio vibrational frequencies are in good
accord with experimental values, whereas the OPLS-AA values are
far off. In Table 2, another example to compare calculated and
observed fundamental frequencies (10 highest modes are shown
here) of trans-N-methylacetamide207 is presented. The ab initio
potential is MP2/DZP and the empirical one is obtained from
AMBER analytical potential for VSCF-PT2 calculation. The parameters from AMBER force-fields are further adjusted in order to
optimize the results, and the experimental frequencies are listed
from matrix experiments.208 Once again it is found that the
ab initio method with no surprise produces much better results
than the other two empirical methods. However, one should
not forget that the force fields such as AMBER, CHARMM and
OPLS-AA were not calibrated for spectroscopy and are generic for
each class of molecule. Thus fitting these force-fields to a certain
molecule or a group of molecules is bound to give better agreement for those properties. Another interesting line of approach
is MD simulation of small biological molecules with less than
100 atoms using direct ab initio methods and it is very much
feasible with present state-of-the-art methods.
However, as stated earlier, it is obvious that direct ab initio
calculations are far more CPU time intensive than the empirical
or semi-empirical methods. Semi-empirical methods, such as
PM3, are computationally very fast than the ab initio methods
and hence feasible for the application to quite large molecules.
They give less satisfactory accuracy compared to ab initio
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Table 2
Summary of VSCF-PT2 frequencies (cm1) for trans-N-methyleacetamide207
Empiricala
Adjusted
empiricalb
Ab initioc
Experimental
Assignment
3309
2986
2985
2984
2985
2872
2868
1676
1598
1459
3310
2985
2986
2985
2986
2873
2869
1662
1569
1462
3523
2993
2985
3014
2979
2940
2939
1751
1547
1566
3498
3008
2978
3008
2973
2958
2915
1708
1511
1472
NH str.
CCH3 asym. str.
NCH3 asym. str.
CCH3 asym. str.
NCH3 asym. str.
NCH3 asym. str.
CCH3 asym. str.
Amide I
Amide II
NCH3 asym. bend
a
AMBER.
b
Adjusted AMBER. c MP2/DZP.
methods, but much better accuracy than the empirical methods.
In such situation use of hybrid potential by Brauer et al.144 is an
attractive choice for the spectroscopic calculations where large
biological molecules like proteins can be treated efficiently. It
has been used successfully for the past few years. Anharmonic
vibrational calculation of proline is one such example. Table 3
shows a few higher frequency vibrational modes of proline using
PM3 and hybrid PM3 (PM3/MP2) potentials. It is found, as
expected, that the hybrid potentials produce much superior
results compared to pure PM3 calculations. That makes this
algorithm very promising to handle even larger biological molecules than say, proline with considerable accuracy. As a further
improvement in potential, Knaanie et al.209 introduced a highly
accurate MP2/MP4 hybrid ab initio potential for the vibrational
spectroscopy calculation of a few small organic molecules
including a maximum of 14 atomic molecule butane. They found
excellent efficiency and accuracy for the new hybrid potential
where the MP4 level of accuracy at the cost of the MP2 level of
calculation is obtained. However, it is still restricted to relatively
small systems (B20 atoms), since MP4 is an extremely CPU time
intensive process even for a harmonic level of calculations.
Among the available ab initio methods, MP2 potentials and
DFT potentials based on standard BLYP and B3LYP functionals
are widely used for the calculation of anharmonic spectra. It is
always tricky to choose a particular method over the others as it
is difficult to judge which potential will work better for a
particular molecule or types of molecules. In a comparative
study by Chaban and Gerber,210 it was found that MP2
Table 3
Proline II frequencies (cm1)144
Expt
PM3
Hybrid (PM3/MP2)
Description
3025
3393
2916
2616
2885
2984
2984
2959
2934
1989
3619
3221
2854
2892
2836
2844
2829
2800
2726
1951
3165
3428
2905
2946
2906
3055
3047
3021
2999
1815
OH str
NH str
CH2 sym str
CH2 sym str
CH2 sym str
CH2 asym str
CH2 asym str
CH2 asym str
CH str
CQO str
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performed better than B3LYP, while BLYP results were often
unsatisfactory for a set of molecules such as allene, propyne,
glycine and imidazole. In contrast, some recent studies by Pele
et al.204 and Šebek et al.211 showed that B3LYP potential energy
surface is superior to MP2 for some hydrocarbon systems.
Thus, at present, there is no ‘‘rule of thumb’’ to choose a
particular ab initio potential over others, and before passing a
judgment, one should verify case by case to get better agreement
with experimental findings.
(b)
Structures: sugars and their hydrides
The contribution of anharmonic effects to the vibrational
spectra is of much interest for carbohydrate chemistry, in
particular sugars, since they have much importance in biochemical processes. The structures of different sugar molecules
are mostly very flexible with the presence of many low energy
conformers, and those structures change considerably from gas
phase to solid phase. The experimental spectra of the solid
crystalline phase are frequently of good resolution. However,
intermolecular interactions due to hydrogen bonding in the
solid are quite strong compared to the gas phase. Consequently,
the lowest energy conformer in the crystal phase is not the same
as in the gas phase. Thus the accurate spectroscopic calculation
of sugars, both in solid and gas phase, may reveal intrinsic
structural and other properties. There is some experimental
evidence existing in the literature for the spectroscopy of
saccharides in the solid phase.212–214 But unfortunately very
few gas phase experimental spectroscopic data of sugars in the
matrix7 have been reported which can be considered as a single
molecule or a few low energy conformers at low temperature in
weak interacting environments.
The anharmonic calculation of sugars has been tested
earlier by Gregurick et al.215 In a recent study, Brauer et al.203
computed anharmonic vibrational spectra for glucose, phenylglucose and sucrose and compared them with experiments in
the gas phase, in an Ar matrix and in the crystalline phase.
A hybrid potential energy surface of the MP2/HF method for
VSCF was used for a-D-glucose and b-D-glucose to compare
with Ar-matrix experimental data of Kovacs and Ivanov.7 Good
agreements were found between the theoretical and experimental frequencies almost throughout the range of data with a
typical error in frequency of 1–2%. They found that intensities
are less satisfactory than the frequencies and inferred that it
might be due to the limitation of the scaling procedure to
construct the hybrid potential.
In an additional test, the comparison of spectroscopic
results of phenyl-b-D-glucose with molecular beam experimental data216 is performed. In Fig. 4, three lowest energy
conformers are shown with experimental and calculated spectra. Theoretical calculations used a computationally fast PM3
method for VSCF-PT2 calculation and also HF/MP2 hybrid
potential to validate PM3 data. Encouraging agreement was
found for both PM3 and HF/MP2 hybrid potential methods.
Most of the measured samples deviated by 1% or even less
showing the supremacy of ab initio methods for such systems.
Additionally, spectroscopy of glucose and sucrose in the crystal
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Fig. 7 Mimic used of the crystalline a-D-glucose. Reprinted with permission from
ref. 203. Copyright 2011, American Chemical Society.
phase was also investigated theoretically using a mimic structure
considering hydrogen bonds with neighboring groups.203 For
example, one molecule in the actual system that is hydrogen
bonded with the ring oxygen of glucose was replaced by methanol.
That greatly simplified the structure for VSCF-PT2 calculation with
PM3 potential. Fig. 7 shows the mimic used for spectroscopy of
a-D-glucose and Fig. 8 shows the crystal state calculation with and
without mimic compared with experiment. Good agreement was
found between computed and experimental spectra in the range
of 500–1500 cm1 especially for the case of mimic.
To assess the temperature effect on sugars, the ab initio MD
was performed using BLYP functional at 50, 150 and 300 K with
satisfactory results. The deviation between VSCF-PT2 and MD
frequencies for that OH stretching ranges up to 56 cm1, but is
mostly smaller. This may be due to inaccuracy of the BLYP
method itself. However, this approach efficiently invokes the
temperature effect on a spectrum, such as broadening of
spectra with the increase in temperature of glucose. Overall
this study opens many possibilities to assess theoretically the
Fig. 8
IR spectra of a-D-glucose in crystal calculations. Data taken from ref. 203.
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structure and dynamics of sugars both in gas and solid phases
with variable temperatures.
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(c)
Different spectra for different conformers
It is obvious that at room temperature several conformers of a
biological molecule are present in an ensemble. Even at low
temperature, there are also finite probabilities for a few lowest
energy conformers to contribute to the bulk properties, especially
for flexible molecules. From the spectroscopic point of view,
the peak position of a particular vibrational mode of different
conformers generally resides in different frequencies and hence
the spectral band broadening type phenomena are strongly
dependent on conformers. Therefore, a thorough theoretical
conformational analysis is a must for a better understanding of
spectral features and corresponding vibrational modes. The
biological building blocks such as amino acids have several low
energy conformers separated by only a few kcal mol1 energy
barriers. For instance, FTIR spectroscopy at low temperature
matrices shows three lowest energy conformers of glycine and
theoretically it is found that these three conformers are within
B1.7 kcal mol1.206 The different intramolecular hydrogen
bonding separates one conformer from the other.
A detailed study by Chaban et al.56 for the calculation of IR
spectroscopy of three lowest energy conformers of glycine
showed a different spectral pattern for each conformer. Fig. 9
shows the comparison of three different spectra of glycine
conformers by the VSCF-PT2 method using MP2/DZP potential.
It can be seen that in the higher frequency range the spectra are
similar for conformers 1 and 3, while conformer 2 is quite different.
This is mostly due to the strong intramolecular hydrogen bonding
present in conformer 2. The O–H bond which participates in
hydrogen bonding with nitrogen is elongated. This corresponds to
vibrational frequencies red shifted by 330–340 cm1, which is in
good agreement with experiment.
Brauer et al.144 examined conformational analysis of a few
more biological molecules such as alanine and proline. It was
found that different experimental techniques showed a different
number of conformers for alanine. Synchrotron radiation photoelectron spectroscopy showed only one conformer whereas
electron diffraction spectroscopy showed two conformers. This
ambiguity about the number of existing low energy conformers
can be resolved by assessing the calculated spectra for different
conformers. To investigate this situation, four lowest energy
conformers with 2 kcal mol1 energy separation were chosen
for the calculation of IR spectra using a PM3/MP2 hybrid
method. The difference in the geometry can be attributed to
differences in hydrogen bonding and conjugation of the carboxyl
group. Comparison of several unassigned experimental spectra
with the theoretical results reveals that there is a high likelihood
that all four conformers may be present. For example, the
unassigned line in the O–H stretching region, the bend region
(1100–1400 cm1) and the torsional region (o600 cm1) are
consistent with the presence of the highest energy conformer
among the four chosen conformers. Proline also has similar
situation like glycine with several conformers that are close in
energy. As proline has a role in determining the protein
secondary structure, its conformational study draws much
attention for both experimental and theoretical perspectives.
Fig. 9 VSCF-PT2 vibrational spectra of three lowest energy glycine conformers at MP2/DZP potential. Reprinted with permission from ref. 56. Copyright 2008,
American Chemical Society.
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Theoretical calculations217,218 showed that as many as 18 proline
conformers exist within B11 kcal mol1 of energy. Here again
different experimental techniques give a different number of
conformers. Theoretically two conformers were studied using
PM3/MP2 hybrid potential for VSCF and VSCF-PT2 with success
for different conformers. One interesting conformational study
was also performed for sugars say, a- and b-D-glucose by Brauer
et al.203 It was found experimentally that the structure of such
conformers varies substantially in the gas phase to the solid
crystalline phase. In crystalline glucose, the hydrogen bonded
interaction with neighboring molecules has a major effect and
hence the conformation of sugars. In the gas phase, three
conformers can be identified for a-D-glucose with respect to
the orientation of the rotation of the CH2OH group. Using
ab initio potential it was found that the energy differences
between the three conformers are extremely low and hence it
is difficult to judge which one is the actual global minima.
A similar situation is also found for b-D-glucose. Thus, it is
reasonable to assume that all the three conformers are trapped
in the matrix experiment with similar probability. Assuming
that, good agreement of the calculated spectra (VSCF-PT2) with
the experiment was found for each conformer that supports the
observation. In another recent study, Pincu et al.219 showed
some interesting conformational and dynamical studies for the
isotopic hydration of cellobiose. They found both isolated and
hydrated cellobiose and lactose units present in highly rigid
structures. The cis conformation was adapted over the trans
conformer by the glycosidic linkage bound by intermolecular
hydrogen bonds at low temperature, and good agreement was
found for theoretical spectra with experiment. However, it was
found surprising that at higher temperature (300 K) the same
conformation was maintained using MD simulation without
suggesting any accessible pathway to a trans conformation. Thus
the spectroscopy of conformers is still an open challenge that
leads to this area much stimulating for both the experimental
and theoretical spectroscopy viewpoint.
(d)
Hydrocarbons at room temperature
Hydrocarbons are of major interest both in organic chemistry
and in biology. They are major components of fossil fuels
produced from the organic remains of living organisms and
hence a primary source of energy. The extremely diverse carbon
skeletons of hydrocarbons are the framework of a variety of
biologically important molecules with several functional
groups attached to it. For example, fats are some of the
biologically important molecules which have regions consisting
of hydrocarbon chains. Thus, hydrocarbons work as a backbone for several biological molecules with different types of
C–H bonds. Arguably, the C–H stretching bond is among the
important vibrational bands in molecular spectroscopy in view
of its abundance in naturally occurring compounds. The high
spectral amplitude of the C–H stretching mode is a signature of
these bands and it is used extensively for mapping of lipids,
sterols, carbohydrates and proteins. It is the dominant probe in
sum-frequency generation examination of aliphatic molecules
at the surface. For example, coherent Raman imaging studies of
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Phys. Chem. Chem. Phys., 2013, 15, 9468--9492
lipophilic molecules frequently making use of only symmetric
CH2 stretching mode at 2845 cm1, made it unique among
different methylene rich molecules. Moreover, the nature and
the peak position of the C–H modes change considerably with
temperature. Thus, a better understanding of C–H stretching
bands in hydrocarbons is needed for non-linear vibrational
modes and it will be discussed later in detail. Pele et al. and
Šebek et al. performed studies for long chain and short chain
hydrocarbons204,220 to assess the Raman and IR spectra for C–H
stretching mode in particular.
For long chain hydrocarbons, the Raman spectra of C–H and
C–D (deuterated) structure bands of dodecane have been calculated
using VSCF and VSCF-DCPT2 algorithms220 compared with liquid
state experiment. The harmonic frequencies and Raman intensities
in the C–H stretching region were calculated at the MP2/CC-PVDZ
level at its global minima. The VSCF anharmonic frequencies were
calculated using a hydride PM3/MP2 method. The comparison of
the calculated spectra with experiment is shown in Fig. 10 for the
non-deuterated dodecane using VSCF and VSCF-DCPT2 methods
respectively. As can be seen, excellent agreement is observed for
both the peak position and intensities. The VSCF-DCPT2 method
leads to red shift for most of the frequencies. However, it was found
that the degeneracy effect does not seem to be very important, at
least for this case. In a second study, IR and Raman spectra of C–H
stretching mode in butane were investigated and compared with
gas-phase experiment. Due to the presence of degenerate states, the
VSCF-DCPT2 algorithm was used. To introduce the temperature
effect, the Lorentzian band with FWHH of 10 cm1 was considered.
Note that this width parameter is empirical and it is introduced
since experimentally the bands are broad. The physical origin of
the broad peaks is due to the presence of a large number of
conformers, and this number is probably due to the (room)
temperature of the experiment. For Raman calculated spectra,
the temperature was set to 295 K to get better agreement and the
intensity expression used is harmonically derived. Frequencies
are the only anharmonic part in it. As can be seen in Fig. 5, the
resulting spectra projected at room temperature are in good
agreement with experiments throughout the range. To construct
the potential the B3LYP as well as MP2 methods are tested.
However, it was found that the B3LYP performed somewhat
better than MP2 particularly for this study.
(e)
Hydrogen bonded complexes of peptides and nucleic acids
Vibrational spectroscopy is a major tool for probing the
potential energy surfaces underlying a weakly bound system.
Both the intermolecular and intramolecular weak interactions
are reflected in different energy ranges of the spectra. Comparison
of the results obtained using ab initio vibrational spectroscopy
with experiment is a test for the adequacy of the electronic
structure methods, for the intermolecular interactions as well as
the coupling between intermolecular and intramolecular degrees
of freedom. Most of these types of interactions are due to
hydrogen bonding which is very common in biological molecules
and they determine several bio-chemical processes. To appraise
this issue, the study of hydrogen-bonded biological molecules
is of great importance and is extensively investigated by several
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Fig. 10
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Comparison of the VSCF spectra of a non-deuterated dodecane isotopomer. Data taken from ref. 220.
groups.57,66,143,221–223 One simple hydrogen bonded complex is
the CH3OH–H2O system studied earlier by Chaban et al.143 and
which found good agreement with theoretical and experimental
frequencies and a qualitative agreement with intensities. The
origin of the qualitative agreement of intensities was predicted
due to inadequacy in the theoretical as well as the experimental
approach since both of them are expected to be less accurate for
intensities than the frequencies. A combined theoretical and
experimental study was performed by Brauer et al.148 for
the vibrational spectroscopy of a complex of two nucleaobases
say, the G C base pair. The computed enolic form of the
equilibrium structure of this complex is shown in Fig. 10,
which has three hydrogen bonds. However, one should keep
in mind that the strength and the pattern of hydrogen bonding
nature of solvated DNA bases, isolated DNA bases and isolated
guanine–cytosine base pairs are different from each other, and
one probable solution to reduce or remove these differences is
by measuring spectra at low temperature in matrices. Thus the
experiment was performed in the gas phase beam expansion
techniques and theoretical spectroscopy calculations used
PM3/RI-MP2 hybrid potential. The third most stable G C
conformer with respect to the RI-MP2/TZVPP level of theory
has been considered due to the availability of the most complete set of experimental vibrational frequencies. Nevertheless
this system is highly anharmonic and needs much attention of
accurate theoretical treatment. However, it was found that only
a limited number of pairs of normal modes have strong mutual
interactions and hence other modes can be treated in an
intrinsic anharmonic way where no coupling is present. It
was found that for hydrogenic stretches in this system the
intrinsic and coupling anharmonicity effects are almost in the
same order of magnitude. However, for the intermolecular
modes, which are essentially involved in hydrogen bonds, the
coupling anharmonicity is significantly more important, for
example, in the CQO stretch. Consequently, these couplings
for the hydrogen bonded modes are very strong and one needs
much intensive theoretical treatment for accurate spectral
analysis.
Gregurick et al.60 investigated a few peptide–water complexes such as di-L-serine–H2O and trialanine in an anti-parallel
b sheet configuration using the VSCF method. They found that
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different peptide–H2O complexes exist corresponding to different
hydrogen bonding sites, and that, however, shift the spectrum up
to 50 cm1 for the fundamental frequencies associated with
peptide modes. Essentially some of these intermolecular modes
suggest effective peptide to water energy transfer. Thus theoretical
exploration of the hydrogen bonded complexes for biological
molecules is essential for a better description of bio-chemical
processes.
However, one should note that for very floppy hydrogen
bonded complexes with soft torsional modes, the standard
VSCF algorithm failed frequently. That failure occurred mainly
due to the normal mode description of potential, which is
inadequate for large amplitude vibrations. In such cases better
representation of the coordinate is needed which has already
been discussed earlier.
(f)
Protonated biological molecules
Recent advances in experimental spectroscopy tools have
provided enormous information on the mechanism underlying
fundamental bio-chemical processes such as enzyme substrate
binding, protein folding, nucleic acid tautomerization, etc. IR
photo-dissociation techniques are particularly applicable to
isolated and micro-solvated protonated peptides, amino acids
and proteins, and provide very useful insight into the dissociation
behavior, preferred protonation sites, etc. Similarly, electrospray
ionization techniques can efficiently ionize biological molecules
and provide unique information on non-covalent bonds of much
importance. OH and NH groups in cluster vibrational spectroscopy can be used to characterize these protonated systems for
bio-chemical processes using the potential energy surfaces underlying them and energy flow between vibrational modes. There are
a few combined theoretical and experimental studies performed
by Gerber’s group and others.150,151,196,205,224 Adesokan et al.150
used a VSCF algorithm to interpret a protonated imidazole
(ImH+)(H2O)nN2 (n = 1, 2). The VSCF method with hybrid PM3/
MP2 potential gives excellent results compared to experiment. In
another recent study,151 they calculated two proton-bound amino
acid wires, GlyGlyH+ and GlyLysH,+ with VSCF-PT2 using PM3/
MP2 hybrid potential. Here the results are shown from their
work in which they found excellent agreement with experimental
data for the anharmonic vibrational frequencies. Within VSCF
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Table 4 GlyLysH+: assignment and comparison of theoretical and experimental
frequencies (cm1)151
PM3
mode
Harmonic
MP2/DZP
VSCF-PT2
hybrid PM3
IRPD
Mode
description
1
2
3
4
5
6
7
3822
3598
3844
3526
3393
3619
3147
3564
3475
3488
3242
3393
3300
3147
3584
3470
3426
3410
3371
3300
3138
OH stretch
NH stretch
NH3+ stretch
NH3+ stretch
NH3+ stretch
NH3+ stretch
NH3+ stretch
approximation, the overall deviation for GlyLysH+ species was only
1.35% (Table 4) from experiment. Particularly, for OH stretching and
NH stretching the observed deviations were 0.56% and 0.14%
respectively. The best result was obtained for NH3+ stretching with
0% deviation. The error occurred in the range of 0 cm1 to
164 cm1. That showed the inherent supremacy of the hybrid
potential for the prediction of vibrational spectra of protonated
biological molecules. For GlyGlyH+ species, the mean deviation is
about 1.4% from the experimental frequencies (Table 5). Similar to
the above observation the best results were obtained for the OH
stretching mode with a deviation of 0.28%. However, in this case the
NH3+ symmetric stretching showed a deviation of 3.2%. In addition
to that, the CH stretching modes showed excellent agreement with
an overall deviation of B1%. Here the error occurred in the range of
18 cm1 to 121 cm1.
These comparisons with available experimental results on
the spectra of the protonated species led to the observation that
anharmonic effects clearly improve the agreement to significant
extent. The ab initio VSCF-PT2 algorithm with hybrid potential is
accurate enough for the exploration of such flexible biological
molecules. This potential makes the VSCF algorithm very fast as
well as reliable and can be applied for considerably larger
protein-bound amino acids and peptides. In addition, it is of
great interest to apply such potential in other approaches such
as MD simulations. However, it is to be noted that quite a few
studies already exist in the literature for protonated biological
molecules and others using MD simulation.157,196,205,224–227
(g)
Assignment and interpretation of CH mode transitions
The assortment of the C–H stretching vibrations is ubiquitous
in biological molecules with a vibrational band in the region
between 2800 and 3100 cm1. The band structure with high
Table 5 GlyGlyH+: assignment and comparison of theoretical and experimental
frequencies (cm1)151
PM3
mode
Harmonic
MP2/DZP
VSCF-PT2
hybrid
PM3
IRPD
Mode
description
1
2
3
4
5
6
7
3839
3785
3544
3531
3191
3151
3285
3574
3562
3261
3296
3045
3009
2958
3584
3584
3372
3400
3045
3000
3042
OH stretch
OH stretch
NH3+ sym stretch
NH sym stretch
CH sym stretch
CH sym stretch
CH sym stretch
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Phys. Chem. Chem. Phys., 2013, 15, 9468--9492
spectral amplitude is a typical signature of this kind of mode
which plays a pivotal role in the detection of several biochemical processes. For example, coherent Raman imaging
studies show only one symmetric stretching of methylene
groups at 2845 cm1 for lipid molecules. However, the link
between the vibrational modes and the C–H stretching band
profile is not well understood. The presence of several CH2 and
CH3 groups in bio-organic molecules results in mixed vibrations at very similar energy ranges and that makes the assignment of the spectra more complicated, mostly due to large
broadening. Thus, unpredictable nature of the interpretation of
the C–H stretching vibrational range on the qualitative modeling of vibrational modes and their mutual coupling in larger
molecules makes it a study of interest both experimentally and
theoretically. The overlap of the symmetric and asymmetric
modes with the overtones, combinatorial modes and Fermi
resonance for methylene rich molecules are very sensitive to
conformational change and environmental factors, and posit a
high challenge for theoretical assessment. A few approaches for
the assignment of the spectral band of the C–H modes were
carried out using a normal mode analysis method and a valance
force field derived from empirical data.228–230 Up to recent
times, the interpretation of Raman231,232 and vibrational
CARS233–235 spectra of such systems strongly relied on the
empirically derived normal mode analysis for band assignments. The limited applicability of the assignment of the C–H
stretching modes using empirical assumption makes it difficult
for an accurate study. Hence, a better approach is needed to
assign accurately such important modes which can provide
desired information with improved analytical power of nonlinear vibrational spectroscopy in the C–H stretching range.
In principle, an ab initio based approach can provide the
desired information, and for C–H stretching modes anharmonic
treatment on a first principles based method can provide much
insight into vibrational calculation. The VSCF method is again a
very suitable tool for this purpose.
Šebek et al.220 analyzed Raman spectra of the dodecane
molecule and its isotopomers as a mimic of a lipid molecule
since the CH2 and CH3 groups are very much abundant in lipid
molecules, and the ratio of these groups for dodecane and for a
lipid is very similar. General experimental evidence shows that
the CH2 asymmetric modes are very sensitive to the environment and if the local environment is more disordered then that
broadens the C–H spectra significantly. On the other hand, CH2
symmetric modes are not that sensitive and hence the spectra
are less broadened. For example, in a liquid sample, the CH2
asymmetric modes generally broaden the spectra two or three
times more than the CH2 symmetric mode. Hence it is reasonable to set the full width at the half-height (FWHH) for the CH2
asymmetric mode more than the CH2 symmetric mode. Applying
that good accord was found with the experimental results of
VSCF spectra with hybrid potential. Fig. 5 shows the comparison
of the Raman spectra for non-deuterated dodecane by VSCF with
experiment, which clearly indicates the excellent agreement. The
overall spectrum shows basically three well resolved peaks. Each
of them is caused particularly by one transition with a very high
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intensity. That corresponds to one of the four mode types: CH2
symmetric and asymmetric, CH3 symmetric and asymmetric.
The Raman intensities of CH3 symmetric stretching are considerably small compared to other modes, and spectral bands of
these modes overlap with more intensive CH2 symmetric stretching. The same observation is also found for D-dodecane. In
Table 6, the comparison between the harmonic and anharmonic
normal mode are shown for the CH2 and CH3 symmetric and
asymmetric modes for dodecane. Both the VSCF and degenerate
VSCF with hybrid potential are found very close to each other
and far off from the harmonic values. Hence this supports the
fact that anharmonic treatment is needed for the interpretation
of such complex spectra over the harmonic approximation. That
leaves the message that the VSCF algorithm can yield a reasonable spectrum for long hydrocarbon with a good agreement with
experimental results. This provides us with the indication that
such calculation is equally possible for any system of this class.
In another work by Pele et al.204 the IR and Raman spectra of
the C–H stretching band are investigated for butane using a
VSCF algorithm. However, the vibrational modes of butane are
not so clearly distinguishable like dodecane. It was found that
the four basic types of modes are always mixed, probably due to
the equal number of CH2 and CH3 groups present in the
dodecane. As a result of the mixing between these groups, the
frequencies corresponding to the mode type cannot be resolved
properly. Fig. 5 shows that the order of the normal modes in the
IR spectra is different from that of the Raman spectra. Because,
in this case the transition visible in the IR spectra is always
invisible in the Raman and vice versa while the energy
Table 6 The vibrational frequencies (cm1) of the CH3 and CH2 stretching
modes for non-deuterated dodecane molecules at different levels of
approximations220
Mode
Type
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
sym
sym
sym
sym
sym
sym
sym
sym
sym
sym
sym
sym
asym
asym
asym
asym
asym
asym
asym
asym
asym
asym
asym
asym
asym
asym
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MP2
harmonic
Imp.
PM3-VSCF
Imp. PM3VSCF_DCPT2
Raman
intensities
3056
3053
3054
3054
3056
3057
3061
3064
3071
3071
3077
3077
3094
3095
3097
3102
3107
3115
3122
3127
3132
3135
3172
3172
3175
3175
2864
2867
2866
2862
2886
2886
2874
2878
2886
2867
2850
2861
2941
2940
2942
2949
2958
2961
2970
2973
2977
2984
2905
2910
2935
2941
2858
2824
2860
2833
2836
2858
2850
2871
2864
2855
2772
2802
2935
2935
2936
2945
2955
2960
2966
2968
2973
2980
2972
2881
2897
2904
1
20
0
1
511
0
19
0
92
1
8
332
372
1
47
0
33
0
39
0
28
0
70
1
52
144
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difference of a nearly degenerate state of the same type of
mode is bigger than the energy difference between the other
types of mode. Moreover, it was found that the CH2 asymmetric
vibrational transitions are visible in a different range of the IR
and Raman spectra and for Raman in particular, it has almost
the same frequency with CH2 symmetric one giving an intensive absorption band for different types of transitions. However,
most of the assignments may differ for different potential and
vibrational methods used for the study. Thus it is very important to choose a ‘‘well equipped’’ method for the assignment of
these modes. We note the same kind of complicated observation for sugars203 where the main discrepancy was found for
C–H stretching modes due to the structural difference adapted
for the theoretical calculation. The actual environment of the
sugar molecule affects the C–H stretching part extensively and
hence broadens the experimental spectra. The mimic group
used in the theoretical calculation cannot represent the actual
environment fully and hence the differences are observed for
the environmentally sensitive C–H stretching modes.
Another interesting alternative to assign such behavior of
different C–H stretching is local mode approximation, which is
widely used for the description of overtone spectral features.
This model was introduced by Henry and Siebrand236 to treat a
system as a set of loosely coupled anharmonic oscillators
localized on individual bonds. In principle all the vibrational
modes in a molecule can couple. However, for a set of normal
modes, the bulk of the vibrational amplitude can be found on a
small set of atoms and hence it can be considered as uncoupled
with others. Investigating only the mode of interest, one can
find much insight into those modes in a simplified way. If the
individual vibration of an atom or a group of atoms does not
match with others then this approximation works very well.
That makes this model simpler to assign the differences between
different modes. This model has been used successfully to assign
the peak position in the overtone spectra and intensities of a wide
variety of systems containing equivalent or non-equivalent C–H
bonds.237–240 Thus careful investigation of this technique may
produce much information for the assignment and corresponding interpretation of different C–H modes, in particular
for biological molecules.
VI. Conclusions, future prospects and open
problems
In this review, we discussed various computational algorithms
for quantitative calculations of the vibrational spectra of biological
molecules. Here all the given examples showed the development
and applicability of those algorithms with a quantitative interpretation of experimental findings. Though several methods are
available, we mainly focused on the VSCF algorithm and its other
variants. Several conclusions can be drawn from the examples
presented here. Firstly, the anharmonic treatment is essential for
biological molecules. The extent of anharmonic contribution may
differ for different transitions. However, at least some of the
fundamental transitions show important anharmonic contribution
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for all the molecules mentioned here. This demonstrates the
importance of the anharmonic effects for biological molecules.
Secondly, within the VSCF framework, several electronic structure
methods seem to play the key role in the accuracy of vibrational
spectroscopy calculations. It is expected that improved electronic
structure methods are the key to higher accuracy and better accord
with experiment though this certainly implies a much increased
computational cost. The present state-of-the-art methods such as
MP2, B3LYP etc. produce good agreement with experiments, but
are computationally limited to small bio-molecules, say, 15–20
atoms. For larger systems, the hybrid potential is a promising
direction. We demonstrated the usefulness of hybrid potential
using a semi-empirical electronic structure method such as PM3
along with other ab initio methods. We showed the supremacy of
this hybrid potential to yield good agreement with solely MP2 or
DFT based potential as well as with experiments. That led to an
effective line of approach to address much larger biological molecules. This algorithm is still at an early stage of development and is
very promising for future research. Further progress in this direction may open up many possibilities to deal with large biological
molecules with 50 or even more number of atoms as an audacious
attempt. Finally, these methodologies with a direct use of ab initio
potentials show very encouraging results for molecules in a strongly
interacting host environment with hydrogen bonds. As we presented here, cleverly chosen small mimic group(s) to model the
effect of a host molecule helped to achieve desired accuracy.
However, more general representation of a condensed phase
environment is also a challenge for future developments. The
general classification and compatibility of mimic groups with
respect to the actual environment is still an open question.
Additionally, we showed some possibilities of assessing temperature and conformational effects on the calculated spectra, which is
a common phenomenon for biological molecules. However, further
developments are needed in this direction. Thus calculation of
vibrational spectroscopy of a biological molecule at ambient temperature and in the presence of solvent is still an open challenge
and it is expected to observe more developments in this issue in the
near future. Though this review is focused exclusively on the
frequency domain spectroscopy, there is also major progress in
time domain spectroscopy. 2D-IR spectroscopy offers such interesting possibilities especially for large molecules in gas as well as
condensed phases. The exciting recent experimental developments241,242 along with the theoretical studies243,244 led to the
impression that this may become a major future of ab initio
spectroscopic studies of biological molecules. Another desirable
future direction is the progress on the spectroscopy of ‘‘soft’’
modes.245 This depends, of course, on experimental progress in
measuring low frequencies.
Acknowledgements
We thank all the present and past members of our group who
were involved in the development of the VSCF method. We
thank Dr J. O. Jung, Dr G. Chaban, Dr S. K. Gregurick,
Dr N. Matsunaga, Dr A. Adesokan, Dr Y. Miller, Dr L. Pele,
Dr J. Šebek, R. Knaanie and Dr B. Brauer in particular for their
9488
Phys. Chem. Chem. Phys., 2013, 15, 9468--9492
contributions in the spectroscopy of biological molecules. TKR
thanks Dr B. Brauer, Dr J. Šebek, R. Knaanie, Dr A. Cohen, Dr S.
Saha, Dr S. Banik and Dr V. Sarkar for their help during writing
this review. We thank Research at the Hebrew University that
was supported by resources of the Saeree K. and Louis P.
Fiedler Chair in chemistry (RBG). TKR also thanks the HU for
a post-doctoral PBC Fellowship.
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