CMST 2(1) 87-100 (1996)
DOI:10.12921/cmst.1996.02.01.87-100
APPLICATION OF EXPLICITLY CORRELATED GAUSSIAN
FUNCTIONS TO L A R G E SCALE CALCULATIONS ON
SMALL ATOMS A N D MOLECULES
Jacek Komasa 1 , Wojciech Cencek 1 , and Jacek Rychlewski 1 , 2
1
Department of Chemistry, Adam Mickiewicz University, Grunwaldzka 6,
60-780 Poznań, Poland
2
P o z n a ń Supercomputing and Networking Center, Wieniawskiego 17/19,
61-712 Poznań, Poland
e-mail: [email protected], [email protected],
[email protected]
1
Abstract
E x p o n e n t i a l l y Correlated Gaussian w a v e functions are applied to variational calculations of the
t o t a l electronic energy of several a few-electron atomic and molecular s y s t e m s . It is s h o w n t h a t
this p o w e r f u l approach enables to obtain extremely accurate results not only for two-electron
s y s t e m s b u t also for three- and four-electron a t o m s and molecules.
2
Introduction
One of the m a j o r tasks of the computational quantum chemistry is to search
for solutions of the Schrodinger equation for atomic and molecular systems. T h e
solution presents a wave function, Ψ, potentially bearing complete description of
the system, and its total energy, E. Unfortunately, the Schrödinger e q u a t i o n is
very c o m p l i c a t e d — i t is composed of various singular and differential o p e r a t o r s
a n d , moreover, contains generally 3 times more variables t h a n particles c o m p o sing t h e s y s t e m . Therefore, in practice, we are not able to find exact analytical
solutions, except for trivial one-electron cases. It is n a t u r a l then looking for simplifications either in the Schrödinger equation itself or in t h e solving process.
T h i s , however, causes t h a t practically every solution of t h e Schrödinger e q u a t i o n
is c o n t a m i n a t e d by an inherent error. T h e main goal of t h e q u a n t u m chemistry,
since t h e first days of its existence, is to develop m e t h o d s reducing this error
as much as possible. Since 1990 our research has aimed at finding very accur a t e solutions of t h e Schrödinger equation for small a few-electron a t o m s a n d
molecules.
T h e quality of t h e wave function can be assessed t h r o u g h t h e energy it
yields—the observable accessible experimentally. T h e wave f u n c t i o n enables also
theoretical d e t e r m i n a t i o n of m a n y other observables which can be c o m p a r e d with
87
m e a s u r a b l e quantities. It supplies an additional source of information a b o u t the
quality of the w a v e function. Still, the energy remains the main criterion of the
quality.
T h e usual o b j e c t i v e of interest is an energy difference, i.e. transition, binding,
interaction energy. However, also an a b s o l u t e energy, especially when k n o w n w i t h
high accuracy, plays i m p o r t a n t role in q u a n t u m chemistry:
• it allows to investigate more subtle energetic effects, i.e. relativistic or rad i a t i v e phenomena, and hence, to p e n e t r a t e deeper and deeper into the
n a t u r e of the m a t t e r ;
• s e r v e s as a benchmark for new computational m e t h o d s ;
• enables correct interpretation of experimental d a t a ;
• s u p p o r t s determination of bulk properties.
T h e r e are t w o basic approaches to the total energy calculations:
• t h e t r a d i t i o n a l m e t h o d — b a s e d on the one-electron a p p r o x i m a t i o n — t h e
Hartree-Fock procedure (HF), optionally followed by the C o n f i g u r a t i o n Interaction m e t h o d (CI);
• the
explicitly correlated wave function m e t h o d — a trial w a v e f u n c t i o n
con-
t a i n s explicitly an interelectron distance variable (e.g. Kołos-Wolniewicz
( K W ) w a v e function), so f a r applied successfully only to 2-electron s y s t e m s and 3-electron a t o m s .
For m a n y small s y s t e m s it is possible to determine an accurate t o t a l e n e r g y
e x p e r i m e n t a l l y , s e t t i n g this way a reference point to the assessment of the e n e r g y
error, ΔE, m a d e in particular c o m p u t a t i o n . T h e following d i a g r a m (Fig. 1) presents t h e dependence of the error in microhartree,
μE H
(1
EH
= 4.3597482 •
1 0 - 1 8 J ) , m a d e with the m e t h o d s mentioned above, on the size of the s y s t e m
e x p r e s s e d by the n u m b e r of electrons.
3
Method
A s y s t e m under consideration is defined by a Hamiltonian,
— a quantum-me-
chanical o p e r a t o r describing the kinetics and all the interactions which shall be
taken into account in the model. T h e general nonrelativistic Hamiltonian of an
n-electron
N-nucleus
molecule can be w r i t t e n down as:
(1)
88
F i g u r e 1: Total energy error, ΔE, obtained from HF (•) and CI (Δ) calculations
and f r o m explicitly correlated calculations with Kołos-Wolniewicz w a v e function
( K W , •). Notice a lack of high accuracy results for s y s t e m s with more t h a n t w o
electrons.
are 3-dimensional vectors pointing at i-th electron and I-th nucleus,
where
respectively, m and e are the m a s s and the charge of the electron, M I and ZI—the
m a s s and the charge of the I-th nucleus.
and
are constants with their usual
m e a n i n g . As the electrons are t h o u s a n d s times lighter and move much f a s t e r t h a n
nuclei, the second term in the Hamiltonian (1) is usually dropped. It c o r r e s p o n d s
to a clamped nuclei model which considers molecule as a rigid skeleton of nuclei
surrounded
by m o v i n g electrons.
The variational method applied
to solve the
Schrödinger equation
(2)
has an i m p o r t a n t f e a t u r e : a l w a y s yields energy, E, g r e a t e r or at m o s t equal to
the e x a c t value, E 0 :
(3)
It allows to search for the w a v e function Ψ using the criterion of g e n e r a t i n g as
low energy as possible. T h e variational principle (3) supplies a scale for a s s e s s i n g
t h e quality of trial w a v e functions.
To obtain the w a v e function in an analytical form we will a p p l y the algebraic
approximation: Ψ is represented as a linear combination of a finite n u m b e r of
known
basis
functions.
(4)
T h e k n o w l e d g e of the basis functions enables to look for the algebraic e x p a n s i o n
89
coefficients instead of a general w a v e function of a completely u n k n o w n s h a p e .
In o t h e r w o r d s , we replace the lack of knowledge of Ψ by the lack of k n o w l e d g e
of
the
set
of
the
coefficients
ck.
4
Basis functions
At this point the choice of the analytical form of the basis f u n c t i o n s ,
becomes a basic m a t t e r . A significant contribution in this field comes f r o m t w o
Polish scientists. Włodzimierz Kołos and Lutosław Wolniewicz applied the b a s i s
f u n c t i o n s in t h e f o r m of two-electron James-Coolidge ( J C ) [1] function w r i t t e n
in elliptical coordinates as
(5)
w h e r e r 1 2 and R describe interelectron and internuclear distances, respectively.
is a real and m,n,j,i and p are integer variational p a r a m e t e r s . T h i s f u n c t i o n
b e l o n g s to the class of the explicitly correlated functions. In 1964 Kołos and
Wolniewicz [2, 3], using the generalized form of the JC wave f u n c t i o n , c o m p u ted a variational energy of molecular hydrogen. Their value w a s lower t h a n the
e x p e r i m e n t a l one, a p p a r e n t l y b r e a k i n g the variational rule. Correctness of their
calculations and even validity of the Schrödinger equation w a s questioned until
f e w y e a r s later the experiment w a s r e v i s e d — i t turned out t h a t the c o m p u t a t i o n s
w e r e correct. It w a s a significant success of theory in connection w i t h the comp u t a t i o n a l technique. T o d a y the Kołos-Wolniewicz w a v e function allows to solve
t h e Schrödinger equation for two-electron two-nucleus s y s t e m s w i t h p r a c t i c a l l y
a r b i t r a r y accuracy. For the last 30 y e a r s it has been s e r v i n g as a reference point
for o t h e r w a v e f u n c t i o n s and m e t h o d s in this area of application. U n f o r t u n a tely, it has never been generalized to larger s y s t e m s . Undoubtedly, the r e s u l t s
by Kołos and Wolniewicz p r o m o t e d the q u a n t u m chemistry to the level of t h e
q u a n t i t a t i v e m e t h o d able to answer detailed questions concerning two-electron
molecules, a n d hence, the chemical bonding.
A n o t h e r , the most common w a y of t r e a t i n g the electron correlation is an
application of the CI w a v e function. It is equivalent to r e g a r d i n g
of Eq. (4)
in the f o r m of d e t e r m i n a n t s built of one-electron molecular functions ( o r b i t a l s ) .
T h e o r e t i c a l l y such an expansion converges to the exact w a v e f u n c t i o n . In practice, h o w e v e r , astronomical n u m b e r of t e r m s including functions w i t h v e r y high
a n g u l a r m o m e n t u m would be required to reach s a t i s f a c t o r y accuracy. T h e CI
w a v e f u n c t i o n , in contrast to the KW w a v e function, does not contain explicitly
the interelectron distance variable, r i j , which is the main reason for the v e r y
slow convergence. For a few-electron s y s t e m s this method yields accuracy, ΔE,
of tens and h u n d r e d s of c m _ 1 ( s e e Table 1).
90
Table 1: Total electronic energy error for chosen a few-electron systems obtained
f r o m the best CI wave functions available in the literature.
In 1989 Enrico Clementi following the example of Hylleraas [11], created an
explicitly correlated version of the CI wave function (H-CI). He introduced the
correlation factor replacing the basis function
by ( l + r i j )
Indeed, compared
to the classic CI the convergence has improved but this approach has turned out
computationally so inconvenient t h a t in 1991 Clementi estimated time required
to obtain accuracy of about 10 c m - 1 for the 3-electron H 3 molecule as 3500
years of c o m p u t a t i o n on an IBM 3090 class machine. He predicted t h a t , with
some optimistic assumptions, this time could be reduced to 10-20 years [7, 12].
At this m o m e n t of development, the q u a n t u m chemistry offered a possibility
of performing either very accurate calculations limited to at most two-electron
systems or calculations on larger systems with much worse accuracy.
In 1991 we s t a r t e d a research on another type of the explicitly correlated
wave function—the Exponentially Correlated Gaussians ( E C G ) . A full definition
of t h e E C G wave function is as follows:
(6)
where
is an operator ensuring proper electron and space s y m m e t r y of the
function, and
is properly chosen spin function. T h e space part of the basis
91
function is defined as
,
(7)
are nonlinear
w h e r e i, j run over all the n electrons, and
a i , k ,b i j , k
and
occur in a
p a r a m e t e r s determined variationally. As we can see the variables
q u a d r a t i c f o r m which is a f e a t u r e of the Gaussian t y p e functions. T h e first t e r m
of the e x p o n e n t is common for one electron functions but the second one contains
distances of all the pairs of electrons and is responsible for correlating the motion
of the electrons. T h e elements mentioned a b o v e compose the n a m e of the w a v e
f u n c t i o n — t h e Exponentially Correlated Gaussian function.
5
T h e wave function optimization
E v e r y basis function
contains, depending on the number of electrons, up to
tens of nonlinear p a r a m e t e r s and there are hundreds or t h o u s a n d s of such basis
f u n c t i o n s . Optimization of the total w a v e function, Ψ, requires a location of the
m i n i m u m of the energy as a function of tens of t h o u s a n d s of p a r a m e t e r s . To d a t e
t h e r e is no universal algorithm solving this t y p e of problem and finding a g l o b a l
m i n i m u m for such a function is unfeasible. We have to accept the f a c t t h a t a
m i n i m u m obtained is merely a local one and put the effort on locating it as low
as possible. It t u r n s out t h a t it brings s a t i s f a c t o r y results.
In our algorithm we optimize (using Powell method [13]) s i m u l t a n e o u s l y only
M p a r a m e t e r s belonging to a single basis function with the other p a r a m e t e r s kept
fixed. W h e n we determine the optimal subset of the p a r a m e t e r s we m o v e the
optimization to the next basis function. A f t e r optimizing all the basis f u n c t i o n s
we close the cycle r e t u r n i n g to the first one (see Fig. 2). E v e r y s t e p and e v e r y
cycle lowers slightly the energy. Of course, the number of cycles is limited by an
e n e r g y lowering criterion or, most frequently, by the cost of the c o m p u t a t i o n s .
Figure 2: Optimization scheme.
92
In the optimization process described above the updating and the diagonalization of the matrices is performed millions of times which makes the calculations
very time consuming. Therefore it is very important to use an optimal code and
machines. T h a n k s to the effort put into our algorithms we managed to diminish
the c o m p u t a t i o n time, from the pessimistic thousands of years predicted fewyears ago by Clementi [7, 12]. to days and weeks. Also the progress in c o m p u t e r
technology allows not only to improve the quality of our results but also perform
calculations unfeasible in the past.
T h e algebraic approximation (4) converts the Schrodinger equation (2) into
the m a t r i x form:
This is a common general symmetric eigenproblem
which is solved in two basic stages:
• m a t r i x elements calculation,
• m a t r i x diagonalization.
T h e latter stage can be very effectively vectorized, especially when the o p e r a t i n g
system supplies with dedicated linear algebra libraries. Nowadays, a diagonalization of a dense 1000 X 1000 matrix makes neither time nor memory problems.
During the optimization of a single basis function only one row and one column
of the
and
matrices are u p d a t e d . Utilization of this fact in our algorithm
was one of the m a j o r steps leading to obtaining good results and performance.
T h e former stage, the matrices build-up, because of the m u t u a l m a t r i x elements independence, can be effectively parallelized. T h e loop running over the
m a t r i x indices comprises the standard object undergoing the parallelization. T h e
following diagram (Fig. 3) presents a practical example of the speedup obtained
when the procedure of filling up the matrices is well parallelized. This particular
picture was obtained from the Atexpert performance monitor on the 16-processor
Cray J916.
Figure 3: Performance speedup as a result of the parallelization.
93
T h e access to the parallel c o m p u t e r s is particularly i m p o r t a n t for e x t e n d i n g
our m e t h o d to l a r g e r and larger s y s t e m s — t h e cost of c o m p u t i n g of one m a t r i x
element g r o w s with the n u m b e r of electrons as a factorial function. R e g a r d i n g
t h a t , for fixed basis set size, the diagonalization cost remains c o n s t a n t , the effect i v e n e s s of the m a t r i x elements computation s t a g e becomes the m o s t i m p o r t a n t
f a c t o r of the whole approach. Massively parallel computers would be t h e m o s t
desired tool for this t y p e of problem, especially when larger a t o m s and molecules a r e of interest. Figure 4 shows how d r a m a t i c a l l y g r o w s the cost of the
c o m p u t a t i o n s when going from two-electron to larger s y s t e m s .
F i g u r e 4: T h e cost of c o m p u t a t i o n g r o w t h with the n u m b e r of electrons.
6
Results
R e s u l t s presented in this section s u m m a r i z e the last few y e a r s of our work on
a l g o r i t h m s and source codes. T h e computer p r o g r a m s applied were created exclusively in our l a b o r a t o r y . T h e calculations were p e r f o r m e d d u r i n g last t w o y e a r s
and s o m e of t h e results were not published yet. All the c o m p u t a t i o n s were done at
A d a m Mickiewicz University or Poznań S u p e r c o m p u t i n g and N e t w o r k i n g Center.
6.1
2-electron s y s t e m s
For 2-electron molecules the ECG functions enable to solve the Schrödinger equation w i t h an accuracy of at least 10 significant figures and, as we h a v e s h o w n
on m a n y e x a m p l e s [14, 15], are, from energetic point of view, equivalent to the
Kołos-Wolniewicz w a v e f u n c t i o n s . Table 2 collects the results for the h y d r o g e n
molecule in its g r o u n d and several excited s t a t e s with various spin and s p a c e
94
Table 2: 2-electron systems.
H 2 , ground s t a t e , R = 1.4011 bohr
95
symmetries. In all the cases the energy obtained from the E C G is the most
a c c u r a t e ever published. T h e same quality of the results have been obtained for
and
T h e choice of the systems presented in the tables proves t h a t
t h e E C G g u a r a n t e e high quality of the results regardless of the spin or space
s y m m e t r y , the charge or the number of nuclei. For the ground s t a t e of H 2 the
E C G functions yield the accuracy of a nanohartree. These are the most a c c u r a t e
calculations on a molecular system (except for the trivial 1-electron case) ever
p e r f o r m e d . In the case of the excited states we estimate the energy error as
0.001 c m - 1 for the triplet a and b states and as 0.01 c m - 1 for the singlet states:
B, C and EF. Similarly, for the
the accuracy is about 0.001 c m - 1 , and for
H e H + a b o u t 0.01 c m - 1 . In Table 2 our results are compared with the other best
variational calculations. Obtaining results of this quality is presently possible on
a personal c o m p u t e r in time of several hours.
6.2
3-electron systems
A m o n g the 3-electron systems we have performed calculations for H 3 , helium
and for lithium a t o m (Table 3). T h e energies obtained are
dimer cation
c o m p u t e d with the accuracy of 7 - 9 significant figures and, except for Li, are
significantly more accurate then any other variational results. H 3 is the molecule
which Clementi group was struggled with. Contrary to their pessimistic predictions we can obtain an accuracy of 1 c m - 1 in several days of c o m p u t a t i o n s on
we estimate t h a t the energy error is less
SGI R 8 0 0 0 / 7 5 type machine. For
then 1 c m - 1 . T h e latter calculations were performed for a wide range of the
bond distance and were followed by rovibrational transition calculations. T h e
c o m p u t e d transitions are on average 0.02 c m - 1 in error when compared with
spectroscopic d a t a [25]. T h e calculations on Li atom are in progress. At present
stage the energy error is ca. 0.001 c m - 1 .
6.3
4-electron systems
T h e accuracy of 7 digits is also attainable for 4-electron systems (Table 4) either
atomic, Be, or molecular, He 2 and LiH. In case of the beryllium a t o m the accuracy of our calculations is about 1 c m - 1 . T h e result has verified an experimental
energy of beryllium. It turned out t h a t the latter was slightly too high. Exceptionally a c c u r a t e are the results for the helium dimer—the energy error is in
t h e range of 0.05 — 0.09 c m - 1 . This is the first time when an analytical wave
function yields such an accurate energy for a 4-electron system. At present, we
are working 011 the whole energy curve which, hopefully, will contribute to the
solution of t h e problem of vibrational stability of the system. T h e calculation
on LiH is in progress. At present stage the estimated energy error is less t h a n
9 cm-1.
96
Table 3: 3-electron systems.
7
Summary
T h e E C G functions have been known in q u a n t u m chemistry for over 30 years.
T h e first results with these functions were published in 1960 [33, 34], However,
accused of slow convergence, E C G were rejected for a long time. As we have
proven recently this opinion was unjustified and today the E C G supply m a n y
best variational results.
97
T h e application of the ECG functions has pointed out t h a t the algebraic
a p p r o x i m a t i o n can yield very good results. For the first time one can obtain
a s a t i s f a c t o r y convergence for s y s t e m s with more than 2 electrons. T h i s is a
t u r n i n g point in the computational q u a n t u m chemistry moving the research on
a few-electron s y s t e m s to a new level of development. Graphically the p r o g r e s s
is represented by the new series of circles ( E C G ) on the following d i a g r a m .
F i g u r e 5: Total energy error, ΔE, obtained from HF (•) and CI (Δ) calculations
and from explicitly correlated calculations with Kołos-Wolniewicz w a v e function
( K W , •). A new set of accurate results (ECG,
) develops along the X axis.
Acknowledgment
T h i s work w a s partially s u p p o r t e d by the Polish C o m m i t t e e for Scientific Research (KBN 8 T 1 1 F 010 08p01 and KBN 2 P303 104 06).
98
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