Nonvariational configuration interaction calculations by local scaling methodToshikatsu Koga, Yoshiaki Yamamoto, and Eduardo V. Ludeña Citation: The Journal of Chemical Physics 94, 3805 (1991); doi: 10.1063/1.459752 View online: http://dx.doi.org/10.1063/1.459752 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/94/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Large scale configuration interaction calculations of linear optical absorption of decacene AIP Conf. Proc. 1538, 271 (2013); 10.1063/1.4810072 Large scale configuration interaction calculations of linear optical absorption of octacene and nonacene AIP Conf. Proc. 1512, 848 (2013); 10.1063/1.4791304 Multi-scale multireference configuration interaction calculations for large systems using localized orbitals:Partition in zones J. Chem. Phys. 137, 104102 (2012); 10.1063/1.4747535 Toward large scale vibrational configuration interaction calculations J. Chem. Phys. 131, 124129 (2009); 10.1063/1.3243862 Ionization potentials of CH2: A comparison of the multiconfigurational spin tensor electron propagatormethod with benchmark full configuration interaction and large scale multireference configurationinteraction calculations J. Chem. Phys. 100, 2947 (1994); 10.1063/1.466437

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

155.247.166.234 On: Sat, 22 Nov 2014 09:03:48

Nonvariational configuration interaction calculations by local scaling method Toshikatsu Koga and Yoshiaki Yamamoto Department of Applied Chemistry, Muroran Institute of Technology, Muroran, Hokkaido 050, Japan

Eduardo V. Ludef\a Centro de Quimica, Instituto Venezolano de Investigaciones Cientificas, Apartado 21827, Caracas J020-A, Venezuela

(Received 30 October 1990; accepted 19 November 1990)

To first order, the Hartree-Fock one-electron density of a given state of an N-electron system is identical to the exact density. For this reason, an accurate wave function must fulfill the necessary condition of yielding a one-electron density similar to the Hartree-Fock density. For a given configuration interaction (CI) wave function with arbitrary parameters, we have used the local scaling method in order to construct a modified CI wave function which associates a one-electron density exactly the same as the Hartree-Fock density. The resultant nonvariational CI wave function is shown to have an energy comparable to that from the variational CI calculation for the ground state of the helium atom.

I. INTRODUCTION

The Hartree-Fock orbitals form a complete set of ortho­normal functions and we can construct from them a com­plete set of N-electron Slater determinants, in terms of which the exact wave function may be expanded. However, Bril­louin's theorem I states that a Hamiltonian matrix element between the Hartree-Fock determinant and any singly sub­stituted (excited) determinant is zero; hence singly substi­tuted terms make no contribution to the wave function to first order in a perturbvation theoretical sense.

As a result, the electron density obtained from the Har­tree-Fock wave function, i.e., the Hartree-Fock density, is identical to the exact density to first order. For thie reason, the Hartree-Fock method is expected to give an· accurate description of the one-electron properties. Conversely, the above fact suggests that the exact wave function or a good approximation to it would be associated to an electron den­sity very similar to the Hartree-Fock density.

In the present paper, we study a nonvariational configu­ration interaction (CI) method in which a CI wave function is constructed so as to have the Hartree-Fock density exact­ly. In the next section, the non variational construction of a CI wave function is presented based on the local scaling transformation at the electron density level. 2 Numerical il­lustrations are given in Sec. III for the ground state of the helium atom. The present nonvariational method is shown to give an energy comparable to that obtained from ordinary CI calculations. Atomic units are used throughout this pa­per.

II. NONVARIATIONAL CONSTRUCTION OF CI WAVE FUNCTIONS

For an N-electron system, let an arbitrary "CI" wave function '110 ( {rj }) be given; it may be regarded as a CI wave functions only in view of its functional form. Although the wave function '110 includes parameters such as CI coeffi­cients, orbital exponents, etc., which (at least the CI coeffi­cients) in the usual CI method are variationally determined

so as to minimize the energy, it has been assumed in the present case that they may be given arbitrary (nonopti­mum) values so long as they are physically acceptable. The given CI wave function '110 has an associated one-electron density Po (r) defined by

(1)

Since the parameters in 'II 0 are nonoptimum, the wave func­tion 'II 0 is generally a poor approximation both to the Har­tree-Fock and the exact wave functions and its associated density po(r) is also far from the Hartree-Fock density PHF ( r) of the system.

The two densities po(r) and PHF (r) may be connected to each other by means of a local scaling transformation expressed as follows:

PHF (r) = J(s/r)po(s), (2a)

where J( sir) is the Jacobian for the variable transformation s = s(r) and guarantees the relation

J(s/r)dr = ds. (2b)

Using spherical polar coordinates r = (r,O) and s = (s,cu), the radial transformation s = s(r,O) is explicitly determined either by the differential equation3

ds(r,O)ldr = (rls)2[pHF (r,O)lpo(s,O)], (3a)

or by the integral equation4

rr rS(r,{l) J/HF (x,0)x

2dx = Jo Po(y,0)y

2dy. (3b)

Once the transformation function s = s( r,O) is determined, the Jacobian is obtained by

J(s/r) =PHF(r,O)lpo[s(r,fi),O]. (4)

Using the local scaling transformation s = s(r) thus es­tablished for the two electron densities, we may then gener­ate a new CI wave function as

'IIPHF({r) = [j~/(s;lrY/2]'IIo({s), (5a)

where

J. Chern. Phys. 94 (5), 1 March 1991 0021-9606/91/053805-03$03.00 © 1991 American Institute of Physics 3805

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

155.247.166.234 On: Sat, 22 Nov 2014 09:03:48

3806 Koga, Yamamoto, and Ludena: Configuration interaction calculations

s;=s(r;), or s;=(s;,o,;) withs;=s(r;.o';). (5b)

Because ofEqs. (2a) and (2b), the generated wave function \II PHF has an associated one-electron density which is exactly equal to the Hartree-Fock density PHF (r). This fact is readi­ly proved as follows:

NJI\II PHF (r,r2,···,rN) 12dr2drN

= N J(s/r) JI\IIO(S,S2, ... ,SN) 12ds2dsN

= J (s/r)po(s)

=PHF (r). (6) Since the local scaling transformation does not change

the structure of the wave function, the generated wave func­tion \II PHF is a CI wave function having the Hartree-Fock density. For the reason given in the previous section, we then expect that the generated wave function \II PHF would be a good approximate wave function in contrast to the initial wave function \II o. This conjecture is numerically verified for the helium atom in the next section.

III. ILLUSTRATIVE APPLICATIONS TO HELIUM ATOM

We apply the proposed method to simple approximate CI wave functions for the helium atom in its ground state. The Hartree-Fock density PHF (r) is taken from Ref. 5, which has been shown5 to reproduce the known Hartree­Fock energy6 EHF = - 2.861680.

The simplest CI wave function for this system will be the Eckart function 7 given by

\110 = (2+2S 2)-1/2[ls(rl )ls'(r2 ) + Is'(r l )ls(r2 )],

Is(r) = (a3 hT) 1/2exp( - ar),

Is'(r) = (/33hr)I/2exp ( -/3r),

S = (lsps') = 8(a/3)3/2/(a + /3)3,

(7)

which includes two exponents a and /3 as parameters. The usual variational calculation gives the minimum energy Emin = - 2.875 661 for a opt = 2.18317 and /3oPt = 1.18853. On the other hand, the results for several sets of randomly given a and /3 are summarized in Table I. From this table, it is evident that the wave function (7) is very poor when the parameters are not optimum. For the parameters listed, all the associated energies E( \II 0) lie above

TABLE I. Nonvariational CI calculations for the helium atom. The initial wave function '110 is given by Eq. (7).

Parameters' a f3 E('IIo) E('II p", ) t.E'

1.5 0.8 - 2.586038 - 2.876 392 0.290354 1.8 1.0 - 2.792 317 - 2.876493 0.084176 2.0 1.0 - 2.840 854 - 2.874900 0.034046 2.2 1.2 - 2.875 461 - 2.876 517 0.001056 2.5 1.5 - 2.776 737 - 2.875373 0.Q98636 3.0 2.0 - 2.211 739 -2.872111 0.660 372

• Arbitrarily given parameters. b Energy improvement.

TABLE II. Nonvariatonal CI calculations for the helium atom. The initial wave function '110 is given by Eq. (8).

Parameters' a f3 c E('IIo) E('II PHF) t.E'

2.0 2.0 -0.1 - 2.682443 - 2.870821 0.188378 1.5 1.5 -0.2 - 2.853 991 - 2.873 337 0.019346 2.0 2.5 -0.2 - 2.592 331 - 2.873 962 0.281631 1.5 2.0 -0.3 - 2.869 864 - 2.875 597 0.005733 2.0 2.5 -0.3 - 2.418 653 - 2.876958 0.458305 1.5 2.0 -0.4 - 2.827 181 - 2.874386 0.047205 2.0 2.5 -0.4 - 2.134449 - 2.872 198 0.737749

• Arbitrarily given parameters. b Energy improvement.

EHF except for the fourth entry. Based on the local scaling method, the initial wave function \110 for each set of random a and /3 values has been numerically transformed to \II PHF

which has the electron density PHF' The energy E(\IIPHF) associated with \lip has been evaluated by the Romberg

HF

integration technique.s We find in Table I that the generated wave function \II improves the energy remarkably,

PHF

though the improvement aE depends on the given param-eters. Indeed, all the E(\IIp ) are below the Hartree-Fock

HF

energy EHF • Moreover, the present nonvariational energies E( \II ) compare quite well with the variational energy

PHF

E min ; in fact, some values of E( \II PH .. ) are lower than E min •

From an inspection of these energy values, we may conclude that the nonvariational wave function \II PHF' which has non­optimum parameters, but is forced to have the Hartree­Fock density, is close to the initial wave function \110 with optimum parameters.

A similar study has been carried out starting from an initial wave function given by

\110 = c l [ Is(r l ) Is(r2)] + c2[2s(r l )2s(r2)],

Is(r) = (a3hT)I/2exp( - ar),

2s(r) = (/35/31T)1/2rexp ( -/3r),

c l = (1 + c2 + 2CS 2)-1/2, c2 = CCI'

S= (lsl2s) = 24(a3/3 5/3)1/2/(a +/3)4,

(8)

where a, /3, and C are independent parameters. As compared to the Eckart wave function, the presence of the parameter C

which governs the mixing ratio of the two configurations lends a new aspect to the wave function (8). A variational calculation gives the best energy Emin = - 2.875 658 for a opt = 1.408 46, /3 opt = 1.836 85, and Copt = - 0.326 36. In Table II, the present results for E(\IIo) and E(\IIPHF) are summarized for several sets of randomly given parameter values. From the comparison of the energies EHF , Emin ,

E(\IIo), and E(\II PHF)' we arrive at similar conclusions as for the Eckart case. The wave functions \110 with nonoptimum parameters are poor when left untouched, but when they are modified to yield the Hartree-Fock density, they are greatly improved. Again, the nonvariational energy E( \II PHF) com­pares favorably with the variational energy Emin • It is impor­tant to notice that although the generated wave functions are

J. Chern. Phys., Vol. 94, No.5, 1 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

155.247.166.234 On: Sat, 22 Nov 2014 09:03:48

Koga, Yamamoto, and Ludena: Configuration interaction calculations 3807

not variational in the sense that neither the parameters nor the density have been variationally optimized, they are prop­erly defined functions in the antisymmetrized N-electron Hilbert space, such that energy expectation values ensuing from these wave functions always remain as upper bounds to the exact energy.

In summary, we have shown for the helium atom that the nonvariational CI wave functions modified to have the Hartree-Fock density possess a quality comparable to the variational CI wave functions, even when the initial wave functions are poor. We note that when the wave functions (7) and (8) with their optimum parameters are modified to yield the Hartree-Fock density, the energies - 2.876 514 and - 2.876 538 are obtained. These energies are lower than the corresponding variational energies Emin by 0.000 853 and 0.000 880, respectively. This result corrobo­rates the general effectiveness of the density constraint for

the improvement of approximate wave functions., We anti­cipate that the present approach would be also successful for other atoms and molecules.

'See, e.g., A. Szabo and N. S. Ostlund, Modern Quantum Chemistry; Intro­duction to Advanced Electronic Structure Theory (Macmillan, New York, 1982), p. 128 If.

2E. S. Kryachko and E. V. Ludeiia, Energy Density Functional Theory of Many-Electron Systems (Kluwer Academic, Dordrecht, 1990), p. 506 If.

31. Z. Petkov, M. V. Stoitsov, and E. S. Kryachko, Int. J. Quantum. Chern. 29, 149 (1986).

"E. S. Kryachko and T. Koga, J. Chern. Phys. 91, 1 J08 (1989). 'T. Koga, Phys. Rev. A 41, 1274 (1990). 6c. Froese Fischer, The Hartree-Fock Method for Atoms (Wiley, New York, 1977), p. 28.

7c. Eckart, Phys. Rev. 36, 878 (1930). "See, e.g., M. J. Maron, NumericalAnalysis (Macmillan, New York, 1982), p. 306 If.

J. Chem. Phys., Vol. 94, No.5, 1 March 1991

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

155.247.166.234 On: Sat, 22 Nov 2014 09:03:48