JOURNAL OF CHEMICAL PHYSICS
VOLUME 115, NUMBER 22
8 DECEMBER 2001
Near-resonant absorption in the time-dependent self-consistent field
and multiconfigurational self-consistent field approximations
Patrick Normana)
Department of Physics and Measurement Technology, Linköping University, SE-581 83 Linköping, Sweden
David M. Bishop
Department of Chemistry, University of Ottawa, Ottawa K1N 6N5, Canada
Hans Jo” rgen Aa. Jensen and Jens Oddershede
Department of Chemistry, University of Southern Denmark, DK-5230 Odense M, Denmark
共Received 13 August 2001; accepted 11 September 2001兲
The linear response function has been derived and implemented in the time-dependent
self-consistent field and multiconfigurational self-consistent field approximations with consideration
made for the finite lifetimes of the electronically excited states. Inclusion of damping terms makes
the response function convergent at all frequencies including near-resonances and resonances.
Applications are the calculations of the electric dipole polarizabilities of hydrogen fluoride,
methane, trans-butadiene, and three push–pull systems. The polarizability is complex with a real
part related to the refractive index and an imaginary part describing linear absorption. The relevance
of linear absorption in nonlinear optics is effectively expressed in terms of figures-of-merit. Such
figures-of-merit have been calculated showing that the nonresonant linear absorption must be
considered when the nonlinear optical quality of a material is to be assessed. © 2001 American
Institute of Physics. 关DOI: 10.1063/1.1415081兴
I. INTRODUCTION
The use of propagator methods in quantum chemistry,
and even more so in physics, has a long and honorable history. An excellent review on polarization propagator methods
in atomic and molecular calculations has been written by
Oddershede et al.1 and as well there is the pioneering book
by Linderberg and Öhrn.2 More recently propagator methods
have been developed and applied, alongside standard electronic structure methods, to routinely calculate high-order
molecular properties at both the electron uncorrelated and
correlated levels. Modern and computationally tractable formulations of the linear response functions have been implemented for nearly all standard electronic schemes: The selfconsistent field 共SCF兲, multiconfigurational SCF 共MCSCF兲,3
second-order Møller–Plesset 共MP2兲,4 second-order polarization propagator 共SOPPA兲,5,6 coupled cluster 共CC兲,7,8 density
functional theory 共DFT兲,9 and four-component SCF10 methods.
The number of molecular properties described by linear
response functions is impressive, and includes among others,
excitation energies, oscillator strengths, electric polarizabilities, magnetizabilities, and nuclear shielding constants.
Common to all these applications and implementations, however, is the exclusion of near-resonant perturbations; the reason being the divergence of the linear response functions at
the excitation energies of the system. From an experimental
point of view, however, this region attracts attention since it
offers a direct way of enhancing an optical response through
dispersion. The situation is often characterized by trading
a兲
Electronic mail: [email protected]
0021-9606/2001/115(22)/10323/12/$18.00
large light scattering response for light absorption, the latter
caused by the closeness of any of the relevant optical frequencies to an electronic transition frequency. One is concerned with finding materials with good figures-of-merit
共FOM兲: Numbers which express scattering efficiency relative
to the tendency to absorb. FOM’s of charge-transfer systems
have recently been measured and analyzed in the context of
few-states-models,11,12 and high FOM’s have also been measured for sulfur-rich compounds.13 Despite the fact that
theory claims to guide experimental material design, standard quantum chemical methods have not been adequately
used to address the issue of figures-of-merit.
Near-resonant absorption can be described in terms of
the imaginary part of the linear polarizability 共just as the
nonlinear absorption can be described by the imaginary part
of the nonlinear polarizability兲. The addition of appropriate
damping terms in the standard perturbation expressions of
polarizabilities and hyperpolarizabilities removes the divergences which would otherwise appear.14 The physical significance of these terms is related to the finite lifetimes of the
excited states of the system. It is shown in the present work
that this approach can be adopted in quantum chemical
propagator methods, thereby defining response functions that
parallel the perturbation sum-over-states 共SOS兲 formulas. We
illustrate this with an implementation of the linear response
function in the time-dependent self-consistent field and multiconfigurational self-consistent field approximations and by
providing numerical data of the linear absorption for a range
of molecules. The former method is also known as the timedependent Hartree–Fock approximation 共TDHF兲 or the random phase approximation 共RPA兲, and the latter as the multiconfigurational random phase approximation 共MCRPA兲.
10323
© 2001 American Institute of Physics
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10324
Norman et al.
J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
Electron correlated values of the linear polarizability are determined for hydrogen fluoride and methane, and uncorrelated values are determined for trans-butadiene and three
push–pull systems. Moreover, the linear absorption is compared with the nonlinear scattering response for some common nonlinear optical processes and the corresponding
FOM’s are found.
cal damping terms in the SOS expressions. The damping
terms in the latter approach represent the finite lifetime of the
excited states, and correspond to line broadening in the absorption spectra. In principle, one damping term ⌫ nm for
each transition from state 兩 n 典 to 兩 m 典 should be introduced.
However, in practice, we are unable to determine all these
damping terms and instead use a common ⌫ for all transitions
⌫⫽1/␶ ,
II. THEORY
A. Perturbation expansions
In the presence of a time-dependent, external, electric
field the molecular polarization is expressed as a perturbation
expansion where the coefficients define the molecular properties known as polarizabilities and hyperpolarizabilities. To
third order this expansion introduces the linear polarizability ␣ (⫺ ␻ ; ␻ ), the first-order hyperpolarizability
␤ (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 ), and the second-order hyperpolarizability
␥ (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 , ␻ 3 )
共4兲
where ␶ is a common lifetime of the excited states. This
approximation may appear unreasonable at first sight, but
one should remember that the value of the polarizability is
significantly affected only by the finite lifetime of the excited
state near the photon energy of the external field.
Having introduced damping terms the SOS expression
for the linear polarizability can be written as
␣ ␣␤ 共 ⫺ ␻ ; ␻ 兲 ⫽ប ⫺1 兺
n
␮ ␣ 共 t 兲 ⫽ ␮ ␣0 ⫹ 兺 ␣ ␣␤ 共 ⫺ ␻ ; ␻ 兲 E ␤␻ e ⫺i ␻ t
⫹
␻
⫹
1
2
兺
␻ ,␻
1
1
⫹
6
␻
2
兺
␻1 ,␻2 ,␻3
␻
␻
␻
␤ ␣␤␥ (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 )E ␤ 1 E ␥ 2 e ⫺i( ␻ 1 ⫹ ␻ 2 )t
␻
␴共 ␻ 兲⫽
4 ␲␻ I
¯␣ 共 ⫺ ␻ ; ␻ 兲 ,
c
共2兲
where c is the speed of light and the isotropic average of the
polarizability has been introduced:
¯␣ ⫽
1
3
兺
i⫽x,y,z
␣ ii .
具 0 兩 ␮ ␤ 兩 n 典具 n 兩 ␮ ␣ 兩 0 典
␻ 0n ⫹ ␻ ⫹i ␥
R
␣ ␣␤
共 ⫺ ␻ ; ␻ 兲 ⫽ប ⫺1 兺
n
⫹
再
冎
共5兲
,
⫺
共 ␻ 0n ⫺ ␻ 兲 2 ⫹ ␥ 2
共 ␻ 0n ⫹ ␻ 兲 2 ⫹ ␥ 2
n
共6兲
具 0 兩 ␮ ␣ 兩 n 典具 n 兩 ␮ ␤ 兩 0 典 共 ␻ 0n ⫺ ␻ 兲
具 0 兩 ␮ ␤ 兩 n 典具 n 兩 ␮ ␣ 兩 0 典 共 ␻ 0n ⫹ ␻ 兲
I
␣ ␣␤
共 ⫺ ␻ ; ␻ 兲 ⫽ប ⫺1 ␥ 兺
再
冎
共7兲
,
具 0 兩 ␮ ␣ 兩 n 典具 n 兩 ␮ ␤ 兩 0 典
共 ␻ 0n ⫺ ␻ 兲 2 ⫹ ␥ 2
具 0 兩 ␮ ␤ 兩 n 典具 n 兩 ␮ ␣ 兩 0 典
共 ␻ 0n ⫹ ␻ 兲 2 ⫹ ␥ 2
冎
共8兲
,
where, for convenience, we have used the half linewidth ␥
⫽⌫/2, ␮ ␣ is the electric dipole moment operator along the
molecular axis ␣ , ប ␻ 0n ⫽E n ⫺E 0 are the electronic excitation energies, and the sums are performed over the manifold
of states of the unperturbed system ( 兩 n 典 ) excluding the
ground state ( 兩 0 典 ). It is immediately seen that in the nonresonant region the imaginary part of ␣ depends linearly on ␥ ,
and that it equals zero in the static limit ( ␻ ⫽0) regardless of
␥ . Furthermore, by using the identity
共3兲
From standard time-dependent perturbation theory, we
obtain a sum-over-states expression for the linear polarizability involving the manifold of excited states of the unperturbed system. Such a formula is applicable only in the nonresonant region, since, when photon energies are in the
proximity of the excitation energies of the system, a perturbation analysis is no longer valid. Thus in the near-resonance
case, two alternative strategies have been used: Either turning to a few-states-model or the inclusion of phenomenologi-
␻ 0n ⫺ ␻ ⫺i ␥
R
I
␣ ␣␤ 共 ⫺ ␻ ; ␻ 兲 ⫽ ␣ ␣␤
共 ⫺ ␻ ; ␻ 兲 ⫹i ␣ ␣␤
共 ⫺␻;␻ 兲,
共1兲
where ␮ ␣0 is the permanent electric dipole moment along the
molecular axis ␣ , E ␻ are the Fourier components of the
perturbing fields, and the sums are performed over both positive and negative frequency ( ␻ ) components. The Einstein
summation convention over the subscripts is assumed both
here and elsewhere. Since the perturbing fields as well as the
molecular polarization are real, we have E ␻ ⫽ 关 E ⫺ ␻ 兴 * and
␣ (⫺ ␻ ; ␻ )⫽ ␣ ( ␻ ;⫺ ␻ ) * and similar relations hold for the
hyperpolarizabilities. The real part of the linear polarizability
␣ R is connected to the refractive index and the imaginary
part ␣ I describes the absorption of light quanta. The linear
absorption cross section equals
具 0 兩 ␮ ␣ 兩 n 典具 n 兩 ␮ ␤ 兩 0 典
or, equivalently
␥ ␣␤␥ ␦ 共 ⫺ ␻ ␴ ; ␻ 1 , ␻ 2 , ␻ 3 兲
⫻E ␤ 1 E ␥ 2 E ␦ 3 e ⫺i( ␻ 1 ⫹ ␻ 2 ⫹ ␻ 3 )t ,
再
冋 再 冎册
lim Im
␥ →0
A
B⫺i ␥
⫽A ␲ ␦ 共 B 兲 ,
共9兲
we see, by comparing with Eqs. 共2兲, 共3兲, and 共8兲, that
lim ␴ 共 ␻ 兲 ⫽
␥ →0
4 ␲ 2␻
3បc
兺n
冋
␦ 共 ␻ 0n ⫺ ␻ 兲
兺
i⫽x,y,z
册
兩 具 0 兩 ␮ i兩 n 典 兩 2 ,
共10兲
and it is clear that the regular oscillator strengths are related
to the infinite lifetime approximation of the absorption as
described by the imaginary part of the linear polarizability.
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J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
Near-resonant absorption in TDSCF
As far as the hyperpolarizabilities are concerned, we restrict ourselves to their real parts in the present work, although, in the resonance region, lifetime broadening terms
should also be added to the SOS formulas for these nonlinear
properties. For instance, the imaginary part of one of the
second-order hyperpolarizabilities describes two-photon absorption in the system. With a reasonable laser detuning,
however, polarizabilities 共as will be explicitly demonstrated
below兲 and hyperpolarizabilities can be accurately determined in the infinite lifetime approximation. We refer to Ref.
15 for an implementation of the cubic response function
without damping terms that allows for a determination of the
nonresonant response in second-order nonlinear processes. In
the present context, it is used to determine the nonlinear
scattering response in the figures-of-merit. For the secondorder hyperpolarizabilities we have considered the classical
orientational average
¯␥ ⫽
1
15
兺i j 共 ␥ ii j j ⫹ ␥ i ji j ⫹ ␥ i j ji 兲 ,
共11兲
where the frequency arguments are implicitly understood.
B. Response theory
In this section we describe how the complex polarizability as given by Eq. 共5兲 can be evaluated in the timedependent multiconfigurational self-consistent field 共TDMCSCF兲 approximation, or, equivalently, in the multiconfigurational random phase approximation 共MCRPA兲. In
its simplest form, the MCSCF wave function contains a
single determinant, and, hence, the uncorrelated RPA method
is also included in the formulation which follows.
The reference state of a system is parameterized by
兩0典⫽
兺n c n兩 ␾ n 典 ,
共12兲
where 兩 ␾ n 典 is a set of spin-adapted, configuration state functions. The MCSCF wave function is optimized against both
orbital rotations and configuration expansion coefficients. In
the presence of a perturbing field the reference state will no
longer be stationary but evolve in accordance with the timedependent Schrödinger equation; the time development of
兩 0 典 is expressed as
兩 0 共 t 兲 典 ⫽exp关 i ␬ 共 t 兲兴 exp关 iS 共 t 兲兴 兩 0 典 ,
共13兲
where exp关i␬(t)兴 is an explicit unitary operator generating
time-dependent variations of the reference state in the orbital
space
␬ 共 t 兲 ⫽ ␬ ␯ q †␯ ⫹ ␬ *
␯ q␯ ,
共14兲
and exp关iS(t)兴 generates variations of the reference state in
the configuration space
S 共 t 兲 ⫽S n R †n ⫹S n* R n .
†
共15兲
Here the q denote pairs of creation and annihilation operators (a r† a s ) that transfer electrons from molecular orbital r to
orbital s (q † is the adjoint of q), and the R † denote configuration state transfer operators (R † is the adjoint of R). The
10325
time-dependent polarization of the system placed in a spatially uniform electric field E(t) can then be written as
具 0 共 t 兲兩 ␮ ␣兩 0 共 t 兲 典 ⫽ 具 0 兩 ␮ ␣兩 0 典 ⫺
冕 具具␮
␻ ⫺i ␻ t
d␻
␣ ; ␮ ␤ 典典 ␻ E ␤ e
⫹•••,
共16兲
from which explicit expressions for the response functions
are obtained by insertion and expansion of Eq. 共13兲 共see Ref.
3 for details兲. The linear response 具具 ␮ ␣ ; ␮ ␤ 典典 ␻ is formally identified with minus the linear polarizability
⫺ ␣ ␣␤ (⫺ ␻ ; ␻ ) in the infinite lifetime approximation, and it
is thus readily applied in the nonresonant region. If, however,
the optical frequency approaches an excitation energy of the
system then the value of the response function diverges.
From Eq. 共5兲 we note that the damping terms needed in this
case can be seen as the addition of a constant imaginary
number i ␥ to the optical frequency ␻ , which, in effect,
projects the residue from the real frequency axis onto the
complex frequency plane. Hence, the corresponding linear
response function takes the form
具具 A;B 典典 ␻ ⫹i ␥ ⫽⫺A [1]† 兵 E [2] ⫺ 共 ␻ ⫹i ␥ 兲 S [2] 其 ⫺1 B [1] , 共17兲
where E [2] and S [2] are the so-called Hessian and overlap
matrices, and A [1] and B [1] are the property gradients corresponding to the components of the polarizability. This expression for the linear response function at a complex frequency parallels those seen in the literature for the so-called
retarded polarization propagator,2 where the positive imaginary term is introduced to ensure that the perturbation vanishes as the time t approaches ⫺⬁. However, the physical
meaning of the imaginary term in Eq. 共17兲 is not the same as
in the retarded propagator. In order to evaluate Eq. 共17兲, we
need to solve the linear response equation
兵 E [2] ⫺ 共 ␻ ⫹i ␥ 兲 S [2] 其 N 共 ␻ 兲 ⫽B [1] ,
共18兲
where, in the computational basis, all matrix quantities except for the linear response vector N( ␻ ) are real. The real
and imaginary parts of the response vector couple via the
addition of the purely imaginary number i ␥ to the optical
frequency. We design a strategy for solving this equation
based on the algorithm already developed for the case without damping16. First we note that with N( ␻ )⫽N R ( ␻ )
⫹iN I ( ␻ ), the coupled equations can be written as
兵 E [2] ⫺ ␻ S [2] 其 N R 共 ␻ 兲 ⫽B [1] ⫺ ␥ S [2] N I 共 ␻ 兲 ,
共19兲
兵 E [2] ⫺ ␻ S [2] 其 N I 共 ␻ 兲 ⫽ ␥ S [2] N R 共 ␻ 兲 .
共20兲
We solve Eq. 共18兲, or equivalently Eqs. 共19兲 and 共20兲, by first
considering the eigenvalue equation
兵 E [2] ⫺ ␻ f S [2] 其 X f ⫽0,
共21兲
where ប ␻ f denote the excitation energies of the system and
X f are the corresponding eigenvectors. The reason for doing
this is connected with the fact that, as ␥ approaches zero, the
form of Eq. 共20兲 becomes identical to the eigenvalue equation. For small values of ␥ therefore, N I ( ␻ ) has to be close
to the null vector for frequencies ␻ far from resonance
whereas it will be close to the eigenvector X f for ␻ ⬇ ␻ f . So,
if the optical frequency of the polarizability in Eq. 共17兲 is
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10326
Norman et al.
J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
near a particular excitation energy ␻ f , we would include the
corresponding state 兩 f 典 in the eigenvalue problem in Eq.
共21兲. The eigenvalue and linear response equations are
solved in a reduced space 共spanned by trial vectors兲 by iteratively extending its dimension until a convergence criterion
2n
is fulfilled. The set of 2n J trial vectors 兵 b i 其 1 J generated by
solving the eigenvalue problem are subsequently used to create a starting point for the response vector in Eq. 共18兲. The
expansion coefficients of the response vector in the first iteration (J⫽0) are given by the explicit inversion of the reduced linear response equation
兵 Ẽ [2] ⫺ 共 ␻ ⫹i ␥ 兲 S̃ [2] 其 a⫽B̃ [1] ,
共22兲
where
T [2]
bj ,
Ẽ [2]
i j ⫽b i E
共23兲
T [2]
S̃ [2]
bj ,
i j ⫽b i S
共24兲
T [1]
.
B̃ [1]
i ⫽b i B
共25兲
We thus obtain
nJ
N RJ 共 ␻ 兲 ⫽
兺
i⫽1
冉冊 冉冊
冉冊 冉冊
bz
R
a 2i⫺1
by
by
R
⫹a 2i
bz
i
nJ
N IJ 共 ␻ 兲 ⫽
兺
i⫽1
bz
I
a 2i⫺1
by
where
sidual
and
T
b 2i
⫽(b y
共26兲
i
bz
I
⫹a 2i
by
i
T
b 2i⫺1
⫽(b z b y ) i
,
共27兲
i
b z ) i , and form the re-
R J 共 ␻ 兲 ⫽B [1] ⫺ 兵 E [2] ⫺ 共 ␻ ⫹i ␥ 兲 S [2] 其 N J 共 ␻ 兲 ,
共28兲
or, equivalently,
R RJ 共 ␻ 兲 ⫽B [1] ⫺E [2] N RJ 共 ␻ 兲 ⫹S [2] 共 ␻ N RJ 共 ␻ 兲 ⫺ ␥ N IJ 共 ␻ 兲兲 ,
共29兲
R IJ 共 ␻ 兲 ⫽⫺E [2] N IJ 共 ␻ 兲 ⫹S [2] 共 ␻ N IJ 共 ␻ 兲 ⫹ ␥ N RJ 共 ␻ 兲兲 .
共30兲
If the norm of the residual vector is less than a threshold
value, our complex linear response equation has been solved.
On the other hand, if this is not the case we increment the
iteration counter (J⫽J⫹1) and solve first
I
共 ␻ 兲,
兵 E [2] ⫺ ␻ S [2] 其 N RJ 共 ␻ 兲 ⫽B [1] ⫺ ␥ S [2] N J⫺1
共31兲
and then
R
共 ␻ 兲,
兵 E [2] ⫺ ␻ S [2] 其 N IJ 共 ␻ 兲 ⫽ ␥ S [2] N J⫺1
共32兲
to a threshold 共accuracy兲 determined by the norm of the residual given in Eq. 共28兲. The reduced space is thereby expanded, i.e., new trial vectors are added to the old ones
which are kept as initial vectors. With the new number of
trial vectors 2n J we return to Eq. 共22兲 and start the next
iteration.
After convergence is obtained, the real and complex
parts of the polarizability are given in terms of the response
vector according to
R
␣ ␣␤
共 ⫺ ␻ ; ␻ 兲 ⫽A [1]† N R 共 ␻ 兲 ,
共33兲
I
␣ ␣␤
共 ⫺ ␻ ; ␻ 兲 ⫽A [1]† N I 共 ␻ 兲 .
共34兲
III. COMPUTATIONAL DETAILS
A selection of molecules has been chosen as a test set to
display the characteristics of near resonant and resonant polarizabilities: hydrogen fluoride 共HF兲 and methane (CH4 ), in
the multiconfigurational self-consistent field approximation
and trans-butadiene 共C4H6 ), push–pull butadiene 共PPB兲,
para-nitroaniline 共PNA兲, and di-cyclopentadienyl-ethyne
共CUM兲 in the self-consistent field approximation.
Throughout we have chosen to present results of frequency dependent properties in the optical region based on
the ubiquitous laser wavelength of 694.5 nm and corresponding to ប ␻ ⫽0.0656 a.u. Multiples of this frequency appear in
nonlinear optical processes and the second and third harmonic generated frequencies of 0.1312 and 0.1968 a.u. are
also pertinent. The excited state broadening ⌫ is chosen to be
one of 0, 500, or 1000 cm⫺1 corresponding to 0, 0.062, and
0.124 eV.
The experimental structures of HF17 and CH4 18,19 have
been used with bond distances of 0.9170 and 1.0858 Å, respectively, whereas theoretically optimized structures have
been adopted for the others. The single and double bond
distances of C4H6 20,21 are 1.4562 and 1.3427 Å, respectively,
and bond distances for PPB,22 PNA,23 and CUM24 are to be
found in Fig. 1.
Results for HF, CH4 , and C4H6 are obtained with
Sadlej’s polarization basis set25 共POL兲. A 4-31G basis set
including polarizing and diffuse functions is used for PPB
and PNA, and, finally, a 6-31G basis set including diffuse
functions is used for CUM. Sadlej’s basis set is optimized
with respect to the atomic static polarizability and should,
hence, be adequate here as far as the real part of ␣ is concerned. The other basis sets employed are likely to introduce
basis set deficiencies of the order of 5% for this property.
The basis set quality needed to describe the excited state
spectra, and thereby also the imaginary part of ␣ , is strongly
dependent on the nature of the excited states in question
共valence or Rydberg兲 but a fair overall quality is to be expected for the low lying states involved here.
Electron correlation is retrieved for HF and CH4 by the
use of complete active space 共CAS兲 wave functions with an
active space corresponding to 2s2p3s3 p3d of Ne, the
united atom limit. Thus for HF the active space is chosen to
be 兵5␴ 3␲ 1␦ 其 while the 兵1␴ 其 orbital is kept inactive, a
choice referring to inclusion of orbitals with an MP2 occupancy greater than 0.8⫻10⫺3 . For CH4 , on the other hand,
the active space becomes 兵 2a 1 1e3t 其 and 兵 1a 1 其 is inactive,
thereby including orbitals with an MP2 occupancy greater
than 0.7⫻10⫺3 .
All calculations have been performed with a locally
modified version of the DALTON program.26
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J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
Near-resonant absorption in TDSCF
10327
FIG. 1. Structures.
IV. RESULTS
A. Linear polarizability
1. Hydrogen fluoride
For hydrogen fluoride there are many references in the
literature concerning ␣ R , and our calculations of this property are not in any way an attempt to compete. Recent theoretical work includes electron correlation at the most sophisticated level, and we refer to Larsen et al.27 for calculations
employing full configuration interaction and to Franke
et al.28 for calculations using an explicitly correlated R12
coupled cluster approach. Our uncorrelated and correlated
values in Table I are within 1% and 5%, respectively, of the
best available values for ␣ R (0;0).
The X 1 ⌺ ⫹ →A 1 ⌸ transition energy is quite strongly
electron correlation dependent, the uncorrelated and correlated values being 11.7 and 10.7 eV, respectively, which in
turn will affect the dispersion in the ultraviolet 共UV兲 region.
The ratio ¯␣ R (⫺ ␻ ; ␻ )/ ¯␣ R (0;0) displays maximum values of
5.1 and 5.2 at the RPA and MCRPA levels of theory, respectively, see Fig. 2. In this figure as well as others the real and
imaginary parts of the polarizability are represented by solid
and dashed lines, respectively. At a detuning of 10%, the real
part of the polarizability is not affected by the finite lifetimes
of the excited states. At a smaller detuning than that, on the
other hand, the infinite lifetime approximation is not reliable.
Another common approximation in optics is the so-called
rotating wave approximation that neglects contributions from
high-frequency terms involving sums rather than differences
of ␻ and ␻ 0n . In this approximation it is readily seen from
Eq. 共7兲 that the real part of the polarizability in a two-state
model has a zero crossing exactly at the excitation energy.
The propagator methods adopted in this work rely on neither
the rotating wave approximation nor on a few-state model
but the characteristic crossing does occur close to the excitation energy since the dispersion gradient of ¯␣ R (⫺ ␻ ; ␻ ) is
large in this region. Between 14.0 to 15.2 eV there are two
excited states of ⌺ ⫹ symmetry and one state of ⌸ symmetry
producing an oscillating behavior of the refractive index.
At the optical frequency corresponding to ␭⫽694.5 nm
or at its second or third harmonic generated frequencies,
there is not much absorption ( ␣ I (⫺ ␻ ; ␻ )⭐0.01 a.u.兲. The
Downloaded 24 Jan 2008 to 130.37.17.12. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
10328
Norman et al.
J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
TABLE I. Polarizability 共a.u.兲 of HFa for dynamic frequencies ␻ 共a.u.兲 and
with an excited state lifetime broadening ⌫ 共cm⫺1 ).b
⌫
0
RPA
␻
c
0
0.0656
0.1968
0.35
0.3936
0.4302
0.45
␣ xx
␣ zz
¯␣
␣ xx
␣ zz
¯␣
␣ xx
␣ zz
¯␣
␣ xx
␣ zz
¯␣
␣ xx
␣ zz
¯␣
␣ xx
␣ zz
¯␣
␣ xx
␣ zz
¯␣
1000.0
MCRPA
RPA
MCRPA
Re
Im
Re
Im
Re
Im
Re
Im
4.5
5.7
4.9
4.5
5.8
4.9
4.8
6.2
5.2
6.2
7.4
6.6
8.1
8.2
8.1
•••
•••
•••
0.2
9.8
3.4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
0
0
0
5.1
6.2
5.5
5.2
6.2
5.5
5.6
6.7
6.0
9.0
8.3
8.7
•••
•••
•••
2.5
10.4
5.1
4.3
11.4
6.7
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
0
0
0
0
0
0
4.5
5.7
4.9
4.5
5.8
4.9
4.8
6.2
5.2
6.2
7.4
6.6
8.1
8.2
8.1
5.6
9.1
6.8
0.3
9.8
3.5
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.01
0.05
0.03
0.04
0.20
0.05
0.15
47.53
0.07
31.71
0.65
0.09
0.46
5.1
6.2
5.5
5.2
6.2
5.5
5.6
6.7
6.0
8.9
8.3
8.7
6.4
9.2
7.4
2.5
10.4
5.1
4.3
11.4
6.7
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.01
0.20
0.04
0.15
65.2
0.06
43.5
0.30
0.10
0.23
0.17
0.13
0.16
a
Values are obtained with the POL basis set.
The lifetime broadening ⌫ equals 2 ␥ . The unit conversion factor between
reciprocal centimeters and electron volts is: 1 eV ⫽ 8065.73 cm⫺1 .
c
The first two excitation energies ប ␻ 0n are 0.4302/0.3936 a.u. and 0.5553/
0.5172 a.u. 共RPA/MCRPA兲.
b
strength of the absorption to the A 1 ⌸ state is enhanced by
about one third at the correlated level and thus, in contradistinction with the refractive index, a quantitative description
of the absorption is more difficult to obtain. In further support of this statement, we note that the absorption to the
manifold of states around 14.5 eV is weaker in comparison
to that of the gap state at the RPA level, but that inclusion of
correlation enhances the absorption cross sections of the two
⌺ ⫹ states so that they become comparable to that of the gap
state, see Fig. 2.
2. Methane
The point group of methane is T d , thus all dipole allowed states are of T symmetry and all components of the
polarizability are equal. In Table II it is seen that the effect of
correlation is small on the real part of the polarizability in the
nonresonant region. The same observation has been made for
␣ R (⫺ ␻ ; ␻ ) when treating electron correlation at the MP2
level.29 Even at a detuning of 10%, correlation amounts to no
greater a correction than 3% owing to the fact that the gap
state energy is insensitive to correlation 共values being 11.06
and 11.02 eV at the RPA and MCRPA levels, respectively兲.
At the same level of detuning, we see that the infinite lifetime approximation is perfectly justified, and we find the
ratio ¯␣ R (⫺ ␻ ; ␻ )/ ¯␣ R (0;0) to have a maximum of 9.7 near
the 1 1 T state. Going beyond the gap state, we find the 2 1 T
state around 12.4 eV appearing as a mere ripple on the
␣ R (⫺ ␻ ; ␻ ) curve in Fig. 3; this is due to a very small transition moment. The four T states in the 13.5–15.5 eV region
lead to large oscillations in the refractive index.
Also ␣ I (⫺ ␻ ; ␻ ) of methane is remarkably insensitive to
electron correlation, the absorption reaches values of 0.07
a.u. for the third harmonic generated frequency ␻ ⫽0.1968.
The resonant absorption of the 1 1 T state is 258.0 and 267.4
a.u. at the RPA and MCRPA levels, respectively, and for the
2 1 T state it is 4.6 and 4.8 a.u., respectively. Large absorption
cross sections are seen for the higher excited T states.
3. Butadiene
The optical spectra of butadiene is strongly dominated
by the 1 1 A g →1 1 B u transition, see Fig. 4, and it is well
known that this transition largely governs both the molecular
linear and nonlinear optical properties. With an assumed lifetime broadening of 1000 cm⫺1 , the absorption 共or ¯␣ I ) peaks
at a value of about 820 a.u., see Table III. The calculated gap
state is positioned at 5.91 eV relative to the ground state,
which is in excellent agreement with the experimental intensity maximum found at 5.92 eV.30 This agreement, obtained
at the electron uncorrelated level, has been frequently reported in the literature, and the issue was recently addressed
by the inclusion of electron correlation in the coupled cluster
approximation including triple excitations 共CC3兲.31 The most
accurate CC3 value for this excitation energy is some 0.3 eV
larger than the uncorrelated value31, and it was concluded
that nuclear motions had to be considered in order to get
agreement with the experimental spectrum. Furthermore, it
was shown that the value of the oscillator strength is somewhat overestimated at the RPA level, so that with proper
inclusion of electron correlation the peak of ␣ I in Fig. 4
would be shifted to a higher energy (⬇0.3 eV兲 and also be
somewhat diminished. The correlation is, however, dynamic
in nature, and we have therefore, not attempted to apply the
MCSCF approach to this case. The calculated absorption
spectrum also shows the two lowest states of A u overlapping
around 6.6 – 6.7 eV and the 2 1 B u state at 7.5 eV, all with
relatively low intensities.
In the nonresonant region, we note that the absorption
depends linearly on the lifetime broadening ⌫, see Table III,
as was pointed out above in connection with Eq. 共8兲. In the
resonant region, the result of varying ⌫ is that the smaller the
⌫ the narrower and stronger is the absorption peak 共the integrated absorption remains constant兲. This is also seen in the
table. These observations are of course general and follow
directly from the form of the SOS expression for ␣ I , see Eq.
共8兲.
Finally turning to the refractive index; in the nonresonant region, the values of the real part of the polarizability
for butadiene, components as well as the average value, are
within 1% of those reported with large basis sets.20 Dispersion indicates ¯␣ R (⫺ ␻ ; ␻ )/ ¯␣ R (0;0) to be a factor of 8.1,
and, with a detuning of 10% from the gap state, the infinite
lifetime approximation inflicts errors in ␣ R (⫺ ␻ ; ␻ ) that are
smaller than 1%.
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J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
Near-resonant absorption in TDSCF
10329
FIG. 2. Real 共solid line兲 and imaginary 共dashed line兲
parts of the averaged polarizability ¯␣ (⫺ ␻ ; ␻ ) of HF at
the 共a兲 RPA and 共b兲 MCRPA level with a lifetime broadening ⌫⫽1000 cm⫺1 .
4. Charge-transfer systems
A class of systems that has attracted attention for molecular design in nonlinear optics is the so-called chargetransfer or push–pull class. Interest stems from the particuTABLE II. Polarizability 共a.u.兲 of CH4 a for dynamic frequencies ␻ 共a.u.兲
and with an excited state lifetime broadening ⌫ 共cm⫺1 ) b.
⌫
0
RPA
␻
c
0
0.0656
0.1312
0.1968
0.30
0.35
0.4050
0.4063
0.42
a
1000.0
MCRPA
Re
Im
Re
Im
15.9
16.1
16.8
18.2
23.6
31.0
482.6
•••
⫺14.7
0
0
0
0
0
0
0
•••
0
15.9
16.2
16.9
18.4
23.9
31.7
•••
⫺431.9
⫺11.8
0
0
0
0
0
0
•••
0
0
RPA
Re
MCRPA
Im
Re
Im
15.9
0.00
15.9
0.00
16.1
0.02
16.2
0.02
16.8
0.04
16.9
0.04
18.2
0.07
18.4
0.07
23.6
0.21
23.9
0.22
30.9
0.57
31.7
0.61
136.0 195.8
12.9 267.4
24.2 258.0 ⫺89.5 199.5
⫺13.5
7.34 ⫺10.9
6.43
All tensor components are equal due to symmetry. Values are obtained with
the POL basis set.
b
See footnotes to Table I.
c
The first two excitation energies ប ␻ 0n are 0.4063/0.4050 a.u. and 0.4560/
0.4539 a.u. 共RPA/MCRPA兲. Both states are of T symmetry.
larly large nonlinear responses that these systems display and
this is connected to localized charge being transported over a
conjugated backbone as the systems undergo electronic transitions. The excited states are often referred to as chargetransfer 共CT兲 states, and they are seen to be highly intensive
in the absorption spectrum. For this reason few-statesmodels have been applied for the evaluation of the linear and
nonlinear optical properties of these systems. It is our experience, however, that with the inclusion of some ten of the
lowest excited states convergence is still not found for
␣ R (0;0) of para-nitroaniline 共PNA兲, and systematic studies
of the frequency dependent polarizability are very difficult in
this approach. Other than PNA, our set of molecules includes
a push-pull polyene 共PPB兲, and di-cyclopentadienyl-ethyne
共CUM兲. The latter is a model system for potential molecular
materials in optical limiting applications.24 Our polarizability
results for these molecules are given in Tables IV–VI.
At the RPA level, the CT states are located at 5.02, 4.78,
and 2.79 eV for PNA, PPB, and CUM, respectively. The
similar gap energies of PNA and PPB are accompanied by
polarizabilities ¯␣ R (0;0) that are similar: 89.5 and 86.5 a.u.,
respectively, whereas CUM, with a gap energy in the visible
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10330
Norman et al.
J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
FIG. 3. Real 共solid line兲 and imaginary 共dashed line兲
parts of the averaged polarizability ¯␣ (⫺ ␻ ; ␻ ) of CH4
at the RPA level with a lifetime broadening ⌫⫽1000
cm⫺1 .
region, has a significantly larger polarizability of 251.0 a.u.
At a laser detuning of 10 %, ¯␣ R (⫺ ␻ ; ␻ )/ ¯␣ R (0;0) amounts
to 1.8, 2.6, and 2.1 for PNA, PPB, and CUM, respectively.
Moreover, with the same detuning, it is seen that errors in
¯␣ R (⫺ ␻ ; ␻ ) within the infinite lifetime approximation can be
as large as 2%.
For these systems, the absorption cross sections are noticeable at the frequency ␻ ⫽0.0656, corresponding to a laser
wavelength ␭⫽694.5 nm; the values of ¯␣ I (⫺ ␻ ; ␻ ) with ⌫
⫽1000 cm⫺1 are 0.29, 0.49, and 5.88 a.u. for this series of
molecules. The second harmonic generated frequency is beyond the CT transition frequency for CUM while the imaginary parts of the polarizability for PNA and PPB at this
frequency are 1.35 and 3.28 a.u., respectively. The third harmonic generated frequency is greater than the CT transition
frequency in all three cases. The resonant CT absorption
cross sections are comparable to that of the ␲ ⫺ ␲ * transition
in butadiene; the ratios with respect to butadiene are 0.7, 1.4,
and 1.3 for PNA, PPB, and CUM, respectively. However,
one thing that separates the CT absorption from the conjugated ␲ ⫺ ␲ * absorption is that it is strongly onedimensional. With ␻ equal to the CT transition frequency,
the ratios of the long 共parallel兲 and short in-plane 共perpendicular兲 polarizability components ␣ I储 (⫺ ␻ ; ␻ )/ ␣⬜I (⫺ ␻ ; ␻ )
are 776.1, 115.1, and 265.1 for PNA, PPB, and CUM, respectively, whereas the corresponding value for butadiene is
11.4. The charge conjugation of orbitals involved in these
transitions is thus enhanced for the molecules in the order
PNA⬍CUM⬍PPB⬍C4 H6 . Such facts may be important for
molecular design.
B. Figures-of-merit
Two simple figures-of-merit are pertinent for the design
of nonlinear optical materials: The ratio of nonlinear scattering to linear absorption and the ratio of nonlinear scattering
to nonlinear absorption. The FOM’s express the fact that not
FIG. 4. Real 共solid line兲 and imaginary 共dashed line兲
parts of the averaged polarizability ¯␣ (⫺ ␻ ; ␻ ) of C4H6
at the RPA level with a lifetime broadening ⌫⫽1000
cm⫺1 .
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J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
Near-resonant absorption in TDSCF
TABLE III. Polarizability 共a.u.兲 of C4H6 a for dynamic frequencies ␻ 共a.u.兲
and with an excited-state lifetime broadening ⌫ 共cm⫺1 ) b.
TABLE IV. Polarizability 共a.u.兲 of PNAa for dynamic frequencies ␻ 共a.u.兲
and with an excited state lifetime broadening ⌫ 共cm⫺1 ) b.
⌫
⌫
0
␻
c
0
0.0656
0.1312
0.1968
0.2170
0.23
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
10331
500.0
0
1000.0
Re
Im
Re
Im
Re
Im
86.4
47.8
37.5
57.3
91.8
48.9
38.4
59.7
116.2
53.1
41.5
70.3
312.5
74.7
50.5
145.9
•••
•••
•••
•••
⫺336.4
21.7
71.5
⫺81.1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
•••
0
0
0
0
86.4
47.8
37.5
57.3
91.8
48.9
38.4
59.7
116.1
53.1
41.5
70.3
311.7
74.6
50.5
145.6
155.5
63.0
58.4
92.3
⫺333.4
22.0
71.4
⫺80.0
0.00
0.00
0.00
0.00
0.20
0.04
0.03
0.09
0.79
0.12
0.08
0.33
14.4
1.36
0.30
5.35
4523
395.9
0.68
1640
34.8
3.21
2.01
13.3
86.4
47.8
37.5
57.3
91.8
48.9
38.4
59.7
116.1
53.1
41.5
70.2
309.3
74.4
50.5
144.7
83.3
56.7
58.3
66.1
⫺324.6
22.8
71.0
⫺76.9
0.00
0.00
0.00
0.00
0.40
0.08
0.06
0.18
1.59
0.24
0.17
0.67
28.5
2.70
0.60
10.6
2262
198.2
1.36
820.7
68.0
6.28
3.95
26.0
␻
0
0.0656
0.1312
0.1660
0.18
0.1844
0.19
a
RPA values with the POL basis set.
See footnotes to Table I.
c
The first two excitation energies ប ␻ 0n are 0.2170 a.u. (1B u state of x, y
dipole symmetry兲 and 0.2401 a.u. (1A u state of z dipole symmetry兲.
b
only must the nonlinear scattering response be large but also
that the absorption losses have to be small if the material is
to be used in the practical world. With this in mind, we have
calculated these FOM’s for the larger systems considered in
this work: C4H6 , PNA, PPB, and CUM. As we have mentioned, this set of molecules is often used to model the characteristic features of systems used in real applications. It may
be noted that there are many different FOM’s in the literature
that incorporate other desirable features of nonlinear optical
共NLO兲 materials.32
In Table VII the real parts of the average second-order
hyperpolarizability ¯␥ R (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 , ␻ 3 ) are given for the
dc electric-field induced Kerr effect 共dc-Kerr兲, the intensity
dependent refractive index 共IDRI兲, electric-field induced
second-harmonic-generation 共ESHG兲, and third-harmonicgeneration 共THG兲 processes. All calculations refer to the
RPA method and, in all cases, the optical frequency ␻ equals
0.0656 a.u. Except for the THG value for C4H6 , the laser
detuning is, therefore, never less than 25% and the infinite
lifetime approximation is quite adequate in these cases. In
the THG calculations the detuning amounts to 9% which
puts a conservative error bar of 10% on ¯␥ R when the finite
lifetimes of the excited states are taken into consideration.
We do not include results for ␥ when any of the optical
frequencies is greater than the molecular gap state transition
frequency.
c
0.20
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
500.0
1000.0
Re
Im
Re
Im
Re
Im
94.1
42.8
131.5
89.5
96.8
43.4
139.6
93.3
107.1
45.3
183.5
112.0
119.9
47.4
323.9
163.7
128.9
46.0
977.3
384.1
•••
•••
•••
•••
138.4
48.5
⫺526.7
⫺113.2
155.2
49.6
⫺99.2
35.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
•••
0
0
0
0
0
0
0
0
94.1
42.8
131.5
89.5
96.8
43.4
139.6
93.2
107.1
45.3
183.5
120.0
119.9
47.4
323.1
163.5
128.8
46.5
923.1
366.1
132.6
47.8
28.9
69.7
138.4
48.5
⫺500.8
⫺104.6
155.0
49.6
⫺98.0
35.5
0.00
0.00
0.00
0.00
0.10
0.02
0.32
0.15
0.29
0.05
1.68
0.68
0.60
0.10
12.8
4.52
0.90
0.95
208.2
70.03
1.05
0.20
3256
1086
1.34
0.13
128.5
43.3
2.94
0.13
18.01
7.03
94.1
42.8
131.5
89.5
96.8
43.4
139.6
93.2
107.1
45.3
183.4
111.9
119.8
47.4
320.8
162.7
128.8
47.1
795.3
323.7
132.5
47.8
104.0
94.8
138.3
48.5
⫺434.2
⫺82.5
154.3
49.6
⫺94.3
36.5
0.00
0.00
0.00
0.00
0.19
0.04
0.63
0.29
0.59
0.10
3.36
1.35
1.20
0.21
25.4
8.94
1.79
1.18
349.8
117.6
2.10
0.39
1630
544.1
2.67
0.26
230.1
77.68
5.72
0.26
35.48
13.82
a
RPA values with 4-31G(p,d) basis set.
See footnotes to Table I.
c
The first two excitation energies ប ␻ 0n are 0.1844 a.u. 共CT 2A 1 state of z
dipole symmetry兲 and 0.1780 a.u. (1B 2 state of y dipole symmetry兲.
b
The basis set requirements are more stringent for the
calculations of accurate ␥ values than for the calculations of
␣ . The static and dynamic values of ¯␥ R for butadiene are in
excellent agreement with those obtained with a large basis
set,20 and the choice of basis sets for PNA and PPB rests on
an extensive basis set investigation performed previously for
PNA.23
The static values of the hyperpolarizability ¯␥ R (0;0,0,0)
increase as 14.81, 26.97, 30.07, and 53.20 (⫻103 a.u.兲 for
C4H6 , PNA, PPB, and CUM, respectively 共see Table VII兲.
These responses are enhanced by dispersion which in turn is
closely connected with the excitation energies to any of the
intense states in the linear absorption spectra; these being the
1B u state for butadiene and the respective CT states for the
other species. It is well known that the smaller the excitation
energy the greater the response, and it is often possible to
establish property relations between these observables.33 The
relevant states have excitation energies that decrease as 5.90,
5.02, 4.78, and 2.79 eV for our series of molecules, and one
Downloaded 24 Jan 2008 to 130.37.17.12. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
10332
Norman et al.
J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
TABLE V. Polarizability 共a.u.兲 of PPBa for dynamic frequencies ␻ 共a.u.兲
and with an excited state lifetime broadening ⌫ 共cm⫺1 ) b.
TABLE VI. Polarizability 共a.u.兲 of CUMa for dynamic frequencies ␻ 共a.u.兲
and with an excited state lifetime broadening ⌫ 共cm⫺1 ) b.
⌫
0
␻
c
0
0.0656
0.1312
0.1582
0.17
0.1757
0.18
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
⌫
500.0
1000.0
0
Re
Im
Re
Im
Re
Im
64.7
37.6
157.2
86.5
66.2
38.1
173.6
92.7
72.3
40.1
280.2
130.9
79.3
41.5
561.2
227.3
90.3
42.5
1504
545.5
•••
•••
•••
•••
66.1
34.8
⫺1774
⫺557.7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
•••
0
0
0
0
64.7
37.6
157.2
86.5
66.2
38.1
173.6
92.7
72.3
40.1
280.1
130.8
79.2
41.5
559.3
226.7
89.8
42.5
1451
527.7
82.6
43.5
382.4
169.5
67.1
41.3
⫺1648
⫺513.2
0.00
0.00
0.00
0.00
0.06
0.02
0.66
0.25
0.19
0.05
4.69
1.64
0.50
0.08
29.8
10.1
2.60
0.13
266.9
89.86
43.5
0.40
7033
2364
4.45
3.16
471.4
159.7
64.7
37.6
157.2
86.5
66.2
38.1
173.6
92.7
72.3
40.1
279.8
130.7
79.2
41.5
553.6
224.8
88.6
42.4
1314
481.6
80.9
43.3
177.3
100.5
69.6
42.3
⫺1354
⫺414.1
0.00
0.00
0.00
0.00
0.11
0.04
1.31
0.49
0.37
0.10
9.37
3.28
1.00
0.16
58.8
20.0
4.73
0.26
479.4
161.5
30.6
0.67
3521
1184
7.57
1.96
785.5
265.0
␻
c
0
0.0656
0.0922
0.1024
0.11
0.12
0.1312
0.1353
a
RPA values with 4-31G(p,d) basis set.
b
See footnotes to Table I.
c
The first two excitation energies ប ␻ 0n are 0.1757 a.u. 共CT 2A ⬘ state of x, z
dipole symmetry兲 and 0.1637 a.u. (1A ⬙ state of y dipole symmetry兲.
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
␣ xx
␣yy
␣ zz
¯␣
500.0
1000.0
Re
Im
Re
Im
Re
Im
539.4
136.3
77.2
251.0
702.6
140.8
78.6
307.3
1320
147.7
80.1
515.9
•••
•••
•••
•••
⫺216.4
147.8
81.5
4.3
538.5
156.1
82.4
259.0
1362
183.4
83.6
543.0
•••
•••
•••
•••
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
•••
0
0
0
0
0
0
0
0
0
0
0
0
•••
•••
•••
•••
539.4
136.3
77.2
251.0
702.3
140.8
78.6
307.2
1311
147.6
80.1
512.9
752.0
148.4
80.9
327.1
⫺195.5
147.8
81.5
11.3
538.6
156.1
82.4
259.0
1345
181.4
83.6
536.7
1900
159.4
84.1
714.4
0.00
0.00
0.00
0.00
8.62
0.18
0.05
2.95
86.1
0.57
0.08
28.9
6491
23.4
0.09
2172
158.8
1.05
0.10
53.3
60.1
1.12
0.11
20.4
149.0
8.06
0.13
52.39
582.5
95.9
0.14
226.2
539.3
136.3
77.2
250.9
701.6
140.8
78.6
307.0
1285
147.5
80.1
504.2
690.1
148.2
80.9
306.4
⫺138.1
148.0
81.5
30.5
538.8
156.0
82.4
259.1
1300
176.8
83.6
520.1
1803
161.2
84.1
682.6
0.00
0.00
0.00
0.00
17.19
0.36
0.10
5.88
166.6
1.13
0.15
56.0
3261
12.3
0.18
1091
299.8
2.03
0.20
100.7
119.0
2.22
0.22
40.5
283.7
13.6
0.26
99.2
670.8
51.2
0.28
240.8
a
RPA values with 6-31⫹⫹G basis set.
See footnotes to Table I.
c
The first two excitation energies ប ␻ 0n are 0.1024 a.u. 共CT 2A ⬘ state of x, y
dipole symmetry兲 and 0.1353 a.u. (3A ⬘ state of x, y dipole symmetry兲.
b
would therefore expect this ordering of the molecules to be
seen when assessing nonlinear optical capabilities. Looking
at ¯␥ R (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 ␻ 3 ) for the various optical processes in
Table VII this is indeed the case. However, turning to the
figure-of-merit ¯␥ R (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 ␻ 3 )/ ¯␣ I (⫺ ␻ ; ␻ ) the situation is different. The superiority of one material over another
is no longer obvious, e.g., the use of PPB in a dc-Kerr-based
device appears inferior to the choice of butadiene, even
though their respective ¯␥ R (⫺ ␻ ; ␻ ,0,0) values would suggest
the opposite. On the other hand, in the ESHG process involving ¯␥ R (⫺2 ␻ ; ␻ , ␻ ,0) the FOM favors PPB over butadiene.
An even more striking example is provided by CUM, which,
with its two donor–acceptor substituted ring units connected
by a cumulenic bridge, displays very large nonlinear responses. Its ¯␥ R (⫺ ␻ ; ␻ ,0,0) value is 327.9⫻103 a.u., which
is more or less an order of magnitude greater than the corresponding values for the other molecules. However, its quality
as a nonlinear optical material is inferior to any of the other
species due to large linear absorption 共the FOM is merely
55.76⫻103 a.u.兲. We point out that this is not an effect of
unreasonable small laser detuning, which in this case is still
as large as 35%. As expected these examples show that the
FOM’s depend strongly on both the process and the molecule. Whereas the dispersion itself is predictable from one
optical process to another, allowing it to be described by
universal formulas,34,35 the FOM’s do not appear to follow
such simple laws.
We would also argue that not only should the nonlinear
optical measurement be affected by the absorption of the
incoming light beam, as discussed above, but it should depend as well on the absorption of the scattered photons of
frequency ␻ ␴ on their way out of the sample. We have therefore also included the figure-of-merit ¯␥ R (⫺ ␻ ␴ ; ␻ 1 , ␻ 2 ␻ 3 )/
¯␣ I (⫺ ␻ ␴ ; ␻ ␴ ) in Table VII. An example is provided by the
absorption of the second harmonic generated photons in the
ESHG process. In this case we note that the stronger absorption for PNA and PPB at ␻ ␴ ⫽2 ␻ gives butadiene a comparable FOM to those of the other two molecules despite its
much lower nonlinear response.
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J. Chem. Phys., Vol. 115, No. 22, 8 December 2001
Near-resonant absorption in TDSCF
TABLE VII. Real part of the average second-order hyperpolarizabilitya (103
a.u.兲 for some nonlinear optical processes at the dynamic frequency ␻
⫽0.0656 a.u., figure-of-merit (103 a.u.兲 relating nonlinear scattering to linear absorption,b and gap energiesc E g 共a.u.兲.
C4 H6
¯␥ R (0;0,0,0)
Eg
¯␥ R (⫺ ␻ ; ␻ ,0,0)
Dispersion
¯␥ R / ¯␣ I (⫺ ␻ ; ␻ )
¯␥ R (⫺ ␻ ; ␻ , ␻ ,⫺ ␻ )
Dispersion
¯␥ R / ¯␣ I (⫺ ␻ ; ␻ )
¯␥ R (⫺2 ␻ ; ␻ , ␻ ,0)
Dispersion
¯␥ R / ¯␣ I (⫺ ␻ ; ␻ )
¯␥ R / ¯␣ I (⫺2 ␻ ;2 ␻ )
¯␥ R (⫺3 ␻ ; ␻ , ␻ , ␻ )
Dispersion
¯␥ R / ¯␣ I (⫺ ␻ ; ␻ )
¯␥ R / ¯␣ I (⫺3 ␻ ;3 ␻ )
PNA
14.81
26.97
0.2170
0.1844
dc-Kerr
17.62
34.29
19%
27%
97.89
118.2
IDRI
21.40
48.06
44%
78%
118.9
165.7
ESHG
26.94
69.62
82%
158%
149.7
240.1
40.21
51.67
THG
92.06
•••
522%
•••
511.4
•••
8.68
•••
PPB
CUM
30.07
0.1757
53.20
0.1024
41.54
38%
84.78
327.9
516%
55.76
67.23
124%
137.2
•••
•••
•••
113.1
276%
230.8
34.48
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
a
RPA values obtained in the infinite lifetime approximation.
The linear absorption ␣ I is determined for ⌫⫽1000 cm⫺1 .
c
Excitation energy for the dominating state in the linear absorption spectra.
b
V. DISCUSSION
Computationally tractable expressions for the evaluation
of the linear response function in the multiconfigurational
self-consistent field approximation have been derived and
implemented. The finite lifetime of the electronically excited
states has been considered and the linear response function is
then convergent in the whole frequency region despite the
presence of resonances or near-resonances. This is achieved
by the incorporation of phenomenological damping factors
that lead to complex response function values. The formulation does not depend on any particular assumptions about the
perturbing fields, which may be time-independent or timedependent, internal or external, magnetic or electric, although the implementation is restricted to perturbations described by one-electron operators. Moreover, the approach
taken is generally applicable to propagator methods which
utilize other electronic structure methods. The indisputable
advantage of the propagator technique is that there is no
truncation of the number of excited states which are taken
into account.
The present work includes applications in terms of the
electric dipole polarizability for which the real part is connected with the refractive index and the imaginary part to the
photon absorption. As a consequence of ignoring nuclear
motions in the response methodology, the absorption spectra
are not vibrationally resolved. Especially for excited states
with very distorted equilibrium structures this may be of importance, and a Franck–Condon analysis may be necessary
in order to obtain accurate absorption cross sections.
Results for our selection of molecules show that with a
10333
laser detuning of at least 10%, the real part of the polarizability can be calculated in the infinite lifetime approximation while maintaining an accuracy of 1% or better for HF,
CH4 , and C4H6 . However, for species with extraordinarily
intensive transitions, such as charge-transfer transitions, errors may amount to about 2% at the same level of detuning.
With a lifetime broadening of 1000 cm⫺1 , dispersion enhances the real part of the polarizability by no more than a
factor of ten for the selected species and in the frequency
regions encompassing the first few resonances. It is seen that,
whereas calculations of the refractive index in the offresonance region are, in most cases, relatively insensitive to
electron correlation, this is not the case for calculations of
absorption cross sections. This is of course a direct consequence of the need for a good description of the excited
states in the latter case. We also note that the imaginary part
of the polarizability depends linearly on the damping factor
in the nonresonant region, and, in the resonant region, different choices of the damping factor affect the height and width
of the absorption peak 共the integrated absorption remains
constant兲.
The present work is significant for the field of molecular
design in nonlinear optics. In order for a material to be technically useful it needs not only to demonstrate large nonlinear responses but also small absorption. This consideration
has been effectively described in terms of simple
figures-of-merit,13 which relate nonlinear light scattering to
linear and nonlinear light absorption. We argue that not only
is the absorption at the operating frequency of a device to be
considered but also the FOM at the resulting, scattered, frequency. When these factors are taken into account, the
judged quality of a material may be changed in ways that
would otherwise have been hard to predict. For instance the
push–pull butadiene molecule, which shows both a greater
nonlinear scattering response and a larger dispersion than
para-nitroaniline, turns out to be the less effective of the two
when absorption is taken into account. It is also shown that
when the comparison of the two molecules is made one
should take into account the nonlinear optical process where
they are to be used. A further consideration to be made is that
of nonlinear absorption; at this moment, we are not able to
address this issue since it involves the imaginary parts of the
nonlinear polarizabilities, but we intend to undertake such
calculations in the near future.
ACKNOWLEDGMENTS
Two of the authors 共P.N. and D.M.B.兲 thank the Natural
Sciences and Engineering Research Council of Canada for
funding and H.J.Aa.J. and J.O. thank the Danish Natural Science Research Council 共Grant No. 9901973兲 for funding.
P.N. acknowledges financial support from a photonics project
run jointly by the Swedish Materiel Administration 共FMV兲
and the Swedish Defense Research Establishment 共FOI兲 as
well as for travel grants from the Swedish Natural Science
Research Council and the Knut and Alice Wallenberg Foundation which allowed part of the work to be carried out during a stay of P.N. in Ottawa. Visits by P.N. and D.M.B. to
Odense were covered by a grant from the Danish Natural
Science Research Foundation.
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10334
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