Electron Correlation Methodology - American Chemical Society

Chapter 1

Explicitly Correlated Basis Functions for Large Molecules Claire C. M. Samson and Wim Klopper

Downloaded by 80.82.77.83 on April 10, 2018 | https://pubs.acs.org Publication Date: March 13, 2007 | doi: 10.1021/bk-2007-0958.ch001

Theoretical Chemistry Group, Debye Institute, Utrecht University, P.O. Box 80052, NL-3508 T B Utrecht, The Netherlands

The MP2-R12 methods are developed further towards applications to spatially extended molecular systems. Firstly, a large auxiliary basis set is employed for the resolution-of -identity approximation (RI approximation, that is, a closure relation) such that smaller standard Gaussian basis sets can be used to expand the wave function. This method yields much better convergence to the limit of a complete basis of second-order Møller-Plesset (MP2) theory than the standard MP2 method using the same correlation-consistent Gaussian basis sets. Secondly, a new correlation factor of the form r exp(-yr ) is investigated, where the Gaussian geminal dampens the linear r term at long interelectronic distances. Many long-range integrals then vanish, depending on the magnitude of the adjustable parameter γ. Finally, a similarity-transformed Hamiltonian is investigated using a correlation function exp(F) similar to the one used for the new MP2-R12 method. 2

12

12

12

© 2007 American Chemical Society

Wilson and Peterson; Electron Correlation Methodology ACS Symposium Series; American Chemical Society: Washington, DC, 2007.

1

2

R12 methods: Wave functions linear in r

12


One of the main bottlenecks that is encountered when attempting to find approximate solutions of the time-independent Schrodinger equation for manyelectron systems is the extremely slow convergence of the computed wave function towards the 'exact' wave function with increasingly large basis sets of atomic orbitals. This major bottleneck arises from the singularity of the Coulomb repulsion at electron-electron coalescence. Figure 1 illustrates the cusp of the He ground-state wave function, which is shown as a function of the angle between the two electrons located on a sphere of radius 0.5 ao.

—

•

0

.

. -2

2

8

-

.

Orf$-

-1

0

1

-

!

-—1

e

12

Figure 1. Coulomb hole of the He ground state. (Reproduced with permission from Molecular Electronic-Structure Theory p. 262. Copyright John Wiley and Sons Limited.)

In the early days of quantum mechanics (1,2\ a drastic improvement of the description of the electron correlation was obtained when terms depending explicitly on the interelectronic distances r^ = Irj-rJ were included into the wave function. Unfortunately, the appearance of such linear i\ terms gives rise to arduous integrals to evaluate, making basically impossible the application of these explicitly correlated methods for atoms and molecules with more than four electrons. During the late 1980's and early 1990's (3,4), however new developments based on this fundamental concept have occurred (i.e., the R12 methods), which have extended the application range of the explicitly correlated methods to small and average-sized molecules. Nevertheless, R12 calculations on very large molecular systems are still computationally very demanding for the following reasons: Firstly, large basis 2


3 sets should be employed to satisfy the RI approximation. Secondly, the linear r term generates a significant number of large two-electron integrals when r is large, although these integrals don't contribute to the energy calculation. B y avoiding the computation of these unnecessary integrals, one should be able to save a considerable amount of computational costs. 12

]2

Formulation of R12 theory


In the notation of second quantization, the R12 wave function is expressed as follows:

*

R

1

2

=

* ia

+

Z

w

+

Z

ijab

+

+

Z

tjkl

Z

+

(

»

ijkabc

where O is the Hartree-Fock determinant, i , j , k, ... denote occupied spin orbitals, and a, b, c, ... denote virtual spin orbitals contained in the finite spin orbital basis. The amplitudes t^, t^,... represent the expansion coefficients of the respective excitations 0 " , 0 * , . . . E a c h exited determinant can be written as a product of annihilation and creation operators as shown for the single excitations, 0;=a;aO.

(2)

+

a / ,ab ... are creation operators and a,, a .. are annihilation operators. The only difference between the R12 expansion of the wave fimction and the conventional configuration-interaction (or coupled-cluster) expansion, is the appearance of a new set of double excitations, Jv

< = Z ( C - C ) w

a

^

o)

where a, p,... denote virtual spin orbitals outside the finite spin orbital basis {cp }. Hence, the union of the two sets of p, q, ... and a, P, ... spin orbitals represents a complete set, p

Z k ( 0 ) f e ( 0 | + Z k ( 0 ) k ( O M P

-

(5)


A n externally contracted M P 2 method The MP2-R12 method can be regarded as an externally contracted M P 2 method with (contracted) double excitations into a complete spin orbital basis. To illustrate this, we shall in the following discuss an externally contracted M P 2 method that comprises contracted double excitations into a subspace {qy}, which has been orthogonalized against the orbital basis {cp }, p

(6)

EC-MP2 ijab

ijkl

\

p'q'

with it

2

= («V 0 ) * r ( ) k K 0 ) « (2)> = (P'q'k.a | k l ) .

(7)

In the following, the latter notation is adopted for simplicity. The only difference between the conventional M P 2 and EC-MP2 wave functions is the appearance of a new set of double excitations with expansion coefficients tJJ,. These double M

excitations are spanned by primitive functions O j contracted through the contraction coefficients (r" - r* ) . If the orthogonal subspace would be the true complementary subspace of the cc-pVnZ basis, the externally contracted MP2 method would be strictly equivalent to the explicitly correlated MP2 method. In Table I, we compare the valence-shell MP2 correlation energies of H 0 obtained from the conventional and externally contracted MP2 methods. The orthogonal subspace used in Table I is spanned by the large basis 0=19sl4p8d6f4g3h2i, H=9s6p4d3f2g. The number of basis functions of the large basis that are (nearly) linearly dependent on the cc-pVnZ basis is drastically increased with the cc-pVnZ basis sets. We observe that the externally contracted M P 2 calculations converge faster to the M P 2 limit. As expected, the energies from the externally contracted M P 2 method lie between the standard 2


5 Table I. Valence-shell M P 2 correlation energy (in m E ) of H 0 h

Basis cc-pVDZ cc-pVTZ cc-p V Q Z cc-pV5Z Large basis Limit

MP2

a)

-201.6 -261.5 -282.8 -291.5 -296.1 -300.5

+ Orthogonal subspace -231.0 -273.0 -287.3 -293.0

2

Linearly dependent I 7 II 117

a) 0= 19sl4p8d6f4g3h2i, H=9s6p4d3f2g


SOURCE: Reproduced with permission from Quantum-Mechanical Prediction of ThermO'Chemical Data by Cioslowski, 2001; p.21. Copyright 2001 Kluwer.)

M P 2 values and the energy obtained in the (uncontracted) large basis. A s the size of the orthogonal subspace increases the computational effort of the M P 2 calculation, the externally contracted M P 2 methods does not appear to be a practicable method. It merely illustrates the explicitly correlated MP2-R12 theory.

Orbital-invariant MP2-R12 method Contrarily to conventional M P 2 theory, the original formulation of M P 2 R12 theory (3,4) did not provide the same results when canonical or localized molecular orbitals were used. Indeed, for calculations on extended molecular systems, unphysical results were obtained when the canonical Hartree-Fock orbitals were rather delocalized (5). In order to circumvent this problem, an orbital-invariant MP2-R12 formulation was introduced in 1991, which is the preferred method since then (6),

(8)

MP2-R12 ijab

ijkl

\

aP

with Z|a)(a| = l-I|P>(p| = l-P-

(9)

The matrix elements needed for the evaluation of the MP2-R12 energy can be computed as follows:


6 212, taken from Ref. 9. c) Extrapolated for 1 -+00, taken from Ref. 9. d) From Ref. 8. e) Using the auxiliary basis 32s24pl8dl5fl2g9h6i.


8 R12 methods augmented with Gaussian geminals By augmenting the linear correlation factor r by a Gaussian geminal of the form exp(-yr ), many two-electron integrals become negligible for large distances between two localized molecular orbitals (Figure 2). The Gaussian functions dampen a large number of integrals that are arising in large molecular complexes. 12


12

9m(l)

10' a.u.) for y=00 a " (crosses), γ=0.05 a ' (dots), γ=0.1 a ' (triangles), γ=1.0 a ~ (squares). The dashed line corresponds to the theoretical number of n /8 integrals and the dotted line represents the number of significant electron-repulsion integrals. 0

2

0

2

2

0

0

4

The aim of this new correlation factor is to associate it with a localization procedure such that it will be possible to predict the vanishing integrals beforehand from the distance between the localized molecular orbitals. This method will allow us to minimize the computational costs considerably since the long-range two-electron integrals represent the major part of the integrals to be computed in a large molecular system. Therefore, this new method in conjunction with auxiliary basis sets for the RI will make it possible to use R12 methods on much larger molecules than is possible today.

Similarity-transformed Hamiltonian The aim of our similarity-transformed Hamiltonian is to improve the computation of the correlation energy of conventional configuration-interaction (CI) calculations. In this framework, the conventional wave function is multiplied by the correlation function (14,15,16) T = exp(F)0,

(13)

with Φ the standard Cl-type expansion (i.e., a linear combination of orbital products) and F a correlation function. Since now the required integrals are


10 available from the newly developed R12-methods, we found it interesting to investigate the following correlation function:

m

m

I

Electron Correlation Methodology - American Chemical Society

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