A configuration-interaction-oriented implementation of the complex coordinate method

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. 46. 365-374 (1993)

A Configuration-Interaction-Oriented Implementation of the Complex Coordinate Method

ALEJANDRO SAENZ AND WOLF WEYRICH Fakultat ftir Chemie, Universitat Konstanz, D-W-7750 Konstanz, Germany

PIOTR FROELICH Department of Quantum Chemistry, Uppsala University, S-75120 Uppsala, Sweden

Abstract

An implementation of the complex coordinate method is demonstrated that exploits a new technique for obtaining the matrix representation of the complex dilated Hamiltonian. The purpose is to make the complex coordinate method applicable together with standard numerical ab initio codes designed for large- scale calculations on many-electron atoms and/or molecules. No complex integrals have to be calculated, and no changes of the standard codes are required even in the common case where the kinetic and potential energy components are not stored separately. Instead, two standard (real) CI calculations are used to generate the dilated (complex) CI matrix representation. The performance of the procedure is demonstrated in the context of the GAMESS program and applied to obtain the resonant structure of the Bethe surface pertinent to the absorption spectrum of the helium atom. 0 1993 John Wiley & Sons, Inc.

1. Introduction

The complex coordinate method has been shown to be a useful tool for calculating atomic resonances (see, e.g., [ 1-41). The discretization of the continuum-induced by this method-allows also the calculation of the generalized oscillator strength density for photoionization spectra [5] and Compton scattering in the distorted-wave Born approximation (DWBA) [6,7]. Shortly after the introduction of the theory of complex coordinates by Aguilar, Balslev, Combes, and Simon [8-111, it was realized that its most economic implementation would consist of a method that does not involve the calculation of nonstandard complex matrix elements. The problem of complex integrals obscures the formal simplicity of the complex coordinate method and has been felt to be a major disadvantage preventing its widespread use.

The so-called direct approach, introduced by Doolen et al. [12], makes use of the fact that the knowledge of the CI matrices of the kinetic and potential energy is sufficient in order to obtain their continuation into the complex plane. However, in standard program codes for CI calculations, the information about the kinetic and potential parts of the matrix elements is not directly accessible. For economy in computer storage requirements, this information is sacrificed at the stage of integral calculation.

0 1993 John Wiley & Sons, Inc. CCC 0020-7608/93/030365- 10

366 SAENZ, WEYRICH, AND FROELlCH

To circumvent this problem, Froelich et al. [13] proposed implementing the continuation numerically by a polynomial fit of values for each matrix element as a function of a real scaling parameter and subsequent insertion of a complex scaling parameter into the polynomial. The authors demonstrated the performance of this procedure in the case of a model Hamiltonian that describes the resonances occurring in the predissociation of a diatomic molecule.

In this paper, we now want to show that it is possible to accomplish the above- mentioned continuation in an analytically correct way with only two standard CI calculations and without any changes in existing programs. In the next section, we describe the method, and in the third section, we report results obtained in a test calculation on the helium atom.

2. The Method

The method of complex coordinates defines a complex scaled Hamiltonian k ( v ) by

k(7) = fi(7.l)kfi-'(q), (1)

f i ( r l ) f G ) = rlff(rlr) (2)

with

and

(3) 7 = p . p.

[It should be noted that the transformation given by Eq. (2) has no influence on a normalization factor.]

Doolen et al. [12] recognized the fact that this transformation leads to the following simple form of the Coulombic Hamiltonian:

H(v) = q-*T + v - ' V , (4)

with T and V being the matrices of kinetic and potential energy, respectively. This is the main relation of the so-called direct approach and avoids the need for complex integrals over the basis functions when the kinetic and potential part of the CI matrix are known.

For cases where T and V are not known separately, Froelich et al. proposed to fit each matrix element Hij by a polynomial in the real scaling factor p to get

To obtain the coefficients clij, one has to calculate the matrix elements for at least n different p-values. The continuation into the complex plane is then performed by evaluating Eq. (5) to complex p-values.

Here we propose an analytically correct separation of the potential and kinetic por- tions of the CI matrix elements. This is possible because scaling of the Hamiltonian is equivalent to the use of inversely scaled basis functions with an unscaled Hamiltonian

IMPLEMENTATION OF THE COMPLEX COORDINATE METHOD 367

[ 141:

(( fi ( 7 140 (r))* I fi ( r ) I fi ( 7 1 40 (r )) = (40 * (r I o(7 - ')fi ( r ) fi- ' (7 - ) 140 (r 1) . (6)

Consider now as an example the case of a basis consisting of real Cartesian Gaussians. For a real scaling factor (7 = p ) , the following relation is valid:

- { p 2 r 2 f i ( p ) d r ) = 40p(r) = P: (Px) ' (P .Y )m(Pz)ne

= P qoL(r) = p *

: + l + m + n 1 m n -6p2r2

:+l+m+n T+l+m+n = P x y z e

f i ' (P )&) 9 (7) where the prime denotes the exclusive scaling of the exponents of the Gaussian. The normalization factor N of the Gaussian cpp(r) (which is independent of the scaling factor p as it was mentioned above) is

(8) - ( + I +m +n)N/ = P P '

Therefore, regarding Eqs. (7) and (8), the correctly scaled and normalized basis function [+,,(.)I is equal to the normalized basis function in which only the exponents are scaled [4L(r)], since

CP; ( r ) - ( ; + I + m + n ) N / i + l + m + n 4,,(r) = N * pP(r ) = P P ' P

= NLpL(r) = +L(r). (9)

The insertion of this result (which is also valid for any normalized basis) to Eq. (6) gives, together with Eq. (4),

( 4;, k ( d I m l 4 ; , I ( . ) > = ( 4 P , k ( . ) I f i ( . ) l + P , Ad) = (4k (r)l O ( P -

= p2tkl -k pvkl. (10)

fi- Y P - )I41 (4)

With only two calculations, in which the exponents of all basisfunctions are scaled with the real factors p1 and p2, respectively, and in which these scaled basis functions are normalized according to the different scaling factors, the matrix elements of the kinetic and the potential energy are accessible on the CI level with the relations

(11) T . . = P*Hi; (P?) - P I H i ; ( P z l )

P?P2 - PlP22 IJ

(This is because all matrix elements T;; are linear combinations of tk l only [and V;; are linear combinations of v k l ] . ) The scaling of the exponents of the basis set presents no problem in the standard ab initio programs. The normalization of the basis functions

368 SAENZ, WEYRICH, AND FROELICH

is usually also done by the ab initio program (or could otherwise easily be achieved by a change of the coefficient). Therefore, the source code does not need to be changed-only the CI matrix elements for the total Hamiltonian have to be available.

Since the procedure operates with the matrix elements of the N-particle Hamiltonian at the CI level, it is obviously possible to use Hartree-Fock orbitals as N-particle basis functions. However, it has to be emphasized that the Hartree-Fock orbitals have to be calculated in a given basis set scaled with either pl or p2 and cannot be changed in the further calculation. The advantage of the introduction of the Hartree-Fock orbitals is a decrease of the required size of the expansion and a reduction of the problem of linear dependencies.

The diagonalization of the resulting complex symmetric eigenvalue problem-simplified by the orthogonality of the Hartree-Fock orbitals-gives the eigenvalues and eigenfunctions of the atomic resonances in the usual way by performing a so-called trajectory calculation.

Using the obtained eigensolutions, it is possible to obtain the generalized oscillator strength density [ 151 for photoabsorption, electron-electron, and photon-electron scattering in the context of the DWBA [6,7] by

1; N [c:]' * Q; - c; [cz], * Q,' * c: Im[ z EJ - E: - E

d f ( E , K ) = E 1 1 d(E/Eh) Eh (KUo>2 IT n=2

(13)

with [df(E, K ) V [ d ( E / E h ) ] the generalized oscillator strength density; E , energy transfer projectile to target; K, momentum transfer projectile to target; N, number of CI states; c:, eigenvector i of the complex scaled CI eigenvalue problem; E:, eigenvalue i of the complex scaled CI eigenvalue problem; Qg, matrix built by the expectation values of the elrikrv operator developed in the CI basis (configurations); Eh, 27.21 eV = 2hcR,; and a0 = s. [We want to emphasize that the factor 2 in Eq. (13) arises from the use of atomic units with e = A = me = 1 and, thus, Eh = 1 Hartree instead of hcR, = 1 Rydberg as the energy unit popular in electron scattering work.]

f i Z

3. Implementation and Test Application

The method presented here has been applied in connection with the quantum chemistry program GAMESS [16], which is a typical standard program for Hartree-Fock, MCSCF, and CI calculations using Cartesian Gaussians.

The GUGA-coded CI matrix elements were extracted, decomposed into the kinetic- and the potential-energy parts, and then continued into the complex plane by the procedure described above. As a test case, we have chosen the helium atom with the aim to analyze its metastable states and the photoionization spectrum.. The-not especially optimized-basis was chosen as a set of even-tempered Gaussians by [i = ai - pf with k running from 1 to M, where M , = 11, M p = 10, a, = ap = 0.0085, and p, = p p = 2.9. The ground-state energy obtained by a restricted Hartree-Fock


calculation was EHF = -2.861617Eh. Out of the 41 Hartree-Fock orbitals, 32 were chosen for a full CI calculation (528 configurations), giving a ground-state energy Ecr = -2.9004048h (cf. the value E = -2.90372Eh obtained by Pekeris [17]).

In Figure 1, part of the spectrum of the complex scaled Hamiltonian ( p = 1.0, 8 varying from 0" to 25") is shown. It contains the bound states with energies lower than -2.OEh, the continuum scattering states, and the resonances. If 8 is varied, the bound states stay fixed on the real energy axis, the continuum scattering states migrate strongly, and the resonances migrate only weakly. As opposed to typical plots of the Hamiltonian scaled only with one angle in this kind of plot, one can see that there is no deviation from the straight line describing the first branch cut, as was reported in the paper by Froelich and Flores-Riveros [MI, but there are two branch cuts describing different channels.

An effect not explained up to now, but attributed to the finite expansion size, is that the different branch cuts do not seem to span exactly the same angle with the real energy axis as one would expect. Also, it should be expected that the trajectories attributed to resonances migrate with an angle of 28-like the continuum states-until they are passed by the correct branch cut. After that they should stay fixed. But it can be seen from the figure that the eigenvalues obtained with different values of 8 are migrating with a smaller angle than 28 long before the branch cut has passed the resonance.

Merely enlarging the scales is required to discover the resonances without any previous knowledge (see Fig. 2). Further enlarging makes it possible to locate the single resonances one by one as demonstrated in the case of the lowest-lying resonance in Figure 3.

\

Y E v

H

-2 0 2 Re(E/Eh)

Figure 1. Spectrum of the complex scaled Hamiltonian for p = 1.0, tl = 0"-25'.


0

'i-

9 1 0 -

I _

' 1 I{ 1 : : t . ' . . I . .. . ; ; : : .

:j 1 ;: i

'ti; :

;i 5 : . : * . .. * . * - . . . . .. . . . . . * .

* . * . . . . . . . f . . *

. . . * . . :* : . ... . . : .

In Table I, the energies-only for p = 1.0-for some resonances are collected and compared with other published values. The agreement is quite good, and in concordance with previous experiences, the lowest-lying resonance is located most easily. When comparing with the other values, one has to bear in mind that we have not optimized the scaling factor p and the basis set for the resonances, whereas the other theoretical values are obtained by special optimization procedures.

Using Eq. (13), the photoionization spectrum for helium has been calculated and plotted in Figure 4 for 8 = 26". The integrals between the Cartesian Gaussians and the e"Vk'-operator were calculated analytically by an algorithm given by Barua and Weyrich [19] which was modified in order to calculate these integrals in the case of complex values of 7.

The resonant lines exhibit the typical Funo shape. In this work, we have chosen one scaling angle for the whole spectrum for simplification reasons. However, one has to stabilize the value of the generalized oscillator strength density in principle for every value of E .

The dependence of the calculated generalized oscillator strength density on the scaling angle 8 is demonstrated in Figure 5. The overall stability in the range from 22" to 30" is quite good. However, from the inlet, it can be seen that the stability is different in three parts of the spectrum. In the case of an energy transfer up to about 2Eh, the magnitude of the relative deviations is small, but due to the oscillations, the scaling angle has to be optimized for small intervals of the energy transfer. In the part of the spectrum beyond 2Eh, the magnitude of the relative deviation is increasing

. . ..: :. . i l . . -

. . . . . . . . v . .

G- I -

* . - 0 - . . . 1 -

+ - 9: - 0 - I _ . .

J : . ..

. . I I . , I I .


I I I X X

X

X X X X

X X X

X X X

X X

X

- -

X X X X

X X

- x x x x x XXX JZ -

I I I

A

6 --. w v

G H

0

M I 0 7

M I 0 7

X (u

I

0 I 0 7

X M

I

Figure 3. Stabilization graph (0-graph) to locate the lowest-lying resonance of helium (the most stable point is indicated by a circle and was located numerically).

TABLE 1. Energies and half-widths of some resonances.

State Source Energy (in Eh) r / 2 (in Eh)

a

b -0.777872 0.002268 -0.77722 0.00205

C -0.77746 0.00253

'W)

d -0.7788 (?IS . 0.00254 (20.55 . a

b -0.621928 -0.6002 I S(2) 0.0020

0.000108 d -

a

e

f -0.6941( %0.55 . 0.0007 (20.15 .

-0.6229 (21.1 . -0.66201 0.000644 -0.693135 0.0006825

'W)

aValues obtained in this work from the stabilization graph (with p = 1.0). bValues published by Froelich and Alexander [20], obtained by the complex coordinate method using

'Values obtained by Froelich et al. [21] with an iterative version of the complex coordinate method. dExperimental values published by Hicks and Comer [22]. eValues calculated by Ho [3] by the complex scaling method. fExperimental values obtained by Madden and Codling [23].

200 Slater geminals.


Lo 0

0 1 2 3 4 5

Energy transfer E/Eh Figure 4. Plot of the calculated photoionization spectrum of helium. The resonances show the typical Fano shape, and the absorption edge is automatically included in the calculation

( p = 1.0, e = 260).

Lo 0

0 1 2 3 4 5

Energy transfer E/Eh Figure 5. Plot of the photoionization spectra obtained with five different scaling angles 8.

The inlet contains the relative deviations from the spectrum obtained with B = 26".

IMPLEMENTATION O F THE COMPLEX COORDINATE METHOD 373

TABLE 11. Comparison of the calculated values of the generalized oscillator strength density with published data.

E a h C d e f g h I

0.00 0.05 0.10 0.15 0.20

0.25 0.30 0.35 0.40 0.45

0.50 1 .oo 1 S O 2.00 2.50

1.8427 1.6692 1.5455 1.444 1 1.3536

1.2642 1.1756 1.0900 1.0111 0.9355

0.8683 0.4742 0.3120 0.1927 0.1249

1.8742 1.7015 1.5759 1.4543 1.3443

1.2477 1.1590 1.0764 1.0005 0.9315

0.8693 0.4755 0.3098 0.1935 0.1272

1.878 1.738

1.262 -

- - -

0.873 0.467 0.281 -

1.89 1.74 -

1.23 -

- - -

0.822 0.416 0.242 -

1.83 1.71 -

1.26 - - - -

0.880 0.468 0.278 -

1.77 1.64 -

1.20 - - -

-

0.836 0.448 0.266 -

- 1.716 1.560

1.306 -

-

1.106

0.95 -

-

0.824 0.458 0.280 0.180

- 1.734 1.606

1.378 -

- 1.186

1.028 -

-

0.894 0.486 0.294 0.191

-

1.702 1.586

1.308 -

-

1.116

0.96 -

-

0.832 0.458 0.280 0.186

E is the energy transfer in E h relative to ionization threshold. "Values obtained in this work (with p = 1.0, 0 = 26"). hValues calculated by Froelich and Flores-Riveros [ 181 by the complex coordinate method. 'Theoretical values obtained by Oldham [24]. dValues calculated by Stewart and Webb [25] with the length formula using hydrogenlike functions for

eSarne source as footnote d, but use of Hartree-Fock continuum states. 'Same as footnote e, but use of the velocity formula. Walues calculated by Daskhan and Ghosh [26] using the method of polarized orbitals and with final

hTheoretical values obtained by Bell and Kingston [27]. 'Experimental values measured by West and Marr [28].

the description of the continuum.

states represented by distorted waves.

in dependence of the scaling angle when the energy transfer increases. However, in this part of the spectrum, the values of the generalized oscillator strength density are most stable for one scaling angle in the whole energy range. In the third part, the resonant one, even the shape of the calculated photoionization spectrum is strongly dependent on the scaling angle. In this part, the scaling angle has to be optimized for every resonance.

Table I1 contains the values of the generalized oscillator strength density in comparison with other published theoretical and experimental data.

4. Discussion and Summary

In this work, we have demonstrated how an analytically correct continuation of the CI matrix elements in the complex plane can be achieved by performing only two standard CI calculations. The method offers the possibility to use any standard ab initio program without any change of the code. Since these programs are highly optimized and system-independent, and since they allow a Hartree-Fock calculation as a starting point of the CI calculation, all these advantages are automatically included in


the dilation procedure. The last fact not only improves the reliability of the results, but it also circumvents the problems of linear dependencies in the matrix diagonalization procedure.

Resonances (energy and wave functions) and generalized oscillator strength densi- ties for absorption processes in the framework of the DWBA are available in a simple way. The only difference between this method and the usual procedures for calculating energies and wave functions for bound states is that a complex symmetric matrix has to be diagonalized instead of a Hermitian matrix, but in both cases, the algorithms are known. Therefore, a maximum of analogy to standard calculations is maintained in our procedure.

We have not optimized the basis set, since our primary aim was the demonstration of the validity of the method. Yet, we have been able to show that reasonable results are available without any previous knowledge about the optimum basis set. A detailed examination of the consequences of changing andlor enlarging the basis set and especially of the scaling parameters is in progress, as well as the calculation of the generalized oscillator strength density with K # 0, which describes the scattering processes within the DWBA.

Bibliography

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[lo] B. Simon, Commun. Math. Phys. 27, 1 (1972). (111 B. Simon, Ann. Math. 97, 247 (1973). [12] G.D. Doolen, J. Nutall, and R. W. Stagat, Phys. Rev. A 10, 1612 (1974). [13] P. Froelich, E. Sangfelt, and B. Weiner, Int. J. Quantum Chem. 31, 823 (1987). [14] E. Brandas and P. Froelich, Phys. Rev. A 16, 2207 (1977). [15] M. Inokuti, Rev. Mod. Phys. 43, 297 (1971). [16] M. W. Schmidt, J.A. Boatz, K. K. Baldridge, S. Kosecki, M. S. Gordon, S.T. Elbert, B. Lam, QCPE

Bull. 7, 115 (1985); M. Dupuis, D. Spangler, and J. J. Wendoloski, NRCC Software Catalog (University of California, Berkeley, CA, 1980).

[17] C.L. Pekeris, Phys. Rev. 115, 1216 (1959). [18] P. Froelich and A. Flores-Riveros, J. Chem. Phys. 86, 2674 (1987). [19] M. Barua and W. Weyrich, Acta Crystallogr. A 42, 257 (1986). [20] P. Froelich and S.A. Alexander, Phys. Rev. A 42, 2550 (1990). [21] P. Froelich, M. Hoghede, and S.A. Alexander, J. Chem. Phys. 91, 1702 (1989). [22] P.J. Hicks and J. Comer, J. Phys. B 8, 1866 (1975). [23] R.P. Madden and K. Codling, Astrophys. J. 141, 364 (1965). [24] J. W.B. Oldham, Phys. Rev. 186, 52 (1969). [25] A.L. Stewart and T.G. Webb, Proc. R. SOC. Lond. 82, 532 (1963). [26] M. Daskhan and A.S. Ghosh, Phys. Rev. A 29, 2251 (1984). [27] L. Bell and A.E. Kingston, Proc. Phys. SOC. Lond. 90, 31 (1967). [28] J. B. West and G.V. Marr, Proc. R. SOC. Lond. Ser. A 349, 397 (1976).

Received July 21, 1992 Accepted for publication October 29, 1992

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A configuration-interaction-oriented implementation of the complex coordinate method