Journal of Molecular Structure (Theochem) ,287 (1993) 28 l-285 0166-1280/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

281

Variational calculations for enclosed quantum systems within soft spheroidal boxes: the case of H, Hz and HeH2+

J.L. Marin *pa, Gerard0 Mufiozb

‘Centro de Investigacidn en Ciencias Bkcas, Universidad de Colima, Apartado Postal 2-1694, 28000 Colima, Mexico bUniversidad Autdnoma Metropolitana-Iztapalapa, Departamento de Fisica, Apartado Postal 55-534, 09340 Mexico City, Mexico

(Received 12 October 1992; accepted 9 March 1993)

Abstract

The direct variational method is used to study some physical properties of H, Hi and HeH*+ enclosed within soft spheroidal boxes. The ground state energy, polarizability, and pressure are calculated for these systems as a function of the size and penetrability of the boxes. The results show good agreement when compared with more sophisticated calculations.

Introduction

The study of enclosed quantum systems has received a great deal of attention in the past few years. Most of the interesting properties of these fascinating systems are directly associated with reduced spatial confinement, such as in excitons in semiconductor crystallites [l] and quantum wells [2], atoms and molecules near surfaces [3,4], atoms and molecules under high pressures [5,6] and quantum ballistic systems [7] to mention just a few. Various theoretical approaches have been used to study these systems, ranging from exact solution of Schriidinger’s equation to perturbation theory, as may be corroborated in the above-mentioned references. Recently we have shown [S-lo] that the variational method is also a suitable tool to study a variety of enclosed quantum systems within impenetrable boundaries of different symmetries.

* Corresponding author.

The aim of the present work is to report the first results of our application of the variational method to study quantum systems confined by penetrable boxes of a general spheroidal shape.

In particular, we treat the cases of H, H$, and HeH2+. For these systems we have calculated the ground state energy as a function of box eccentricity for different values of the barrier height. Furthermore, in order to demonstrate the applicability of the method, we have calculated other properties of physical interest directly involv- ing the wavefunction, such as the polarizability and the pressure. The results obtained are in good agreement with other calculations performed with more sophisticated approaches.

H, Hl, and HeH2+ within penetrable spberoidal boxes

The hamiltonian for a one-electron molecular ion in the Born-Oppenheimer approximation can

282 J.L. Marin and G. h4ufioz/J. Mol. Struct. (Theo&m) 287 (1993) 281-285

be written as

H = - 40’ - &/rl - ZJr2 + Z,Z2/2R (1)

where the units are chosen to make A = e = m = 1. The subscripts 1 and 2 denote the nucleus of charges Z1 and Z2 respectively.

The prolate spheroidal coordinates (5, n, ‘p) are defined as [l l]

S = (~1 + r2)/2R n = (II - r2)P cp (2)

where 2R is the interfocal distance of the prolate spheroids of revolution {t = constant; - 1 <n 6 1; O<cpG2r}.

If the nuclei are located in the foci of this coordinate system, the hamiltonian of Eq. (1) can be now expressed as

H = - +V’ - Z,/R(t + q) - ZJR(< - Q)

(3) + Z1 Z2/2R

where the laplacian operator is then

1 v2 = R2(,52 _ $) (c2 - ’

x [(I -#)$]} + l -$ R2(t2 - 1)(1 - $) &’

(4)

We now consider the system as confined within a prolate spheroidal box of eccentricity l/Es and barrier height Vs. The corresponding modified hamiltonian is then

H, = -;V’+ v,(<,n)

where

(5)

v,=

{

-ZilR(<+ 77) - Z2/R(I - 77) (1 Gg50)

b (EoG<G~)

(6)

The formal solutions of Schrodinger’s equation for the inner (Es co) and outer (52 lo) regions must satisfy continuity conditions at the boundary

(< = co) which are equivalent to

(7)

Equation (7) allows us to find the energy of the system if an explicit form of the wavefunction is known.

The V. = 00 case was studied exactly by Ley-koo and Cruz [12]. An approximate study of these systems for finite values of V. can be performed using a direct variational method. In this case an ansatz wavefunction x must be constructed. Following Refs. 9 and 10, we have

x0(5,77, ‘p) = Ao?&G 77, cp; o)f(& o) @b)

where &’ and & are the solutions of Schrodinger’s equation for the free system [ 13,14],fis an auxiliary function which allows the total wavefunction to satisfy the condition shown in Eq. (7) and Q is a variational parameter to be determined once the corresponding energy functional is minimized with the additional constrictions improved by

Eq. (7). Following Coulson [ 131, the ansatz wavefunction

for the ground state is constructed as

xi& 77, ‘P) = MO - 4) evb(S + 41 (<GO)

Pa)

and

x0(6 rl, 9) = A0 ew(-PC) exd-4 (EXO)

P)

where Aj and A0 are two constants related through the normalization condition on the total wave- function. A relation between the variational parameters a and /3 can be obtained through the continuity condition at 5 = co so that the energy functional is to be minimized with respect to only one parameter, either cx or /3.

J.L. Marin and G. MwiozlJ. Mol. Strut. (Theo&em) 287 (1993) 281-285 283

2.25

1.50

-0.75

(4

vo= m

20

8 CRY)

20

8 CRY)

1 2 3 4

(b)

2.05

0.00

& B 5

-2.05

-4.1( 1 -

vo= Ccl

20

8 CRY) 2

0

HeH++

IS0

@I ’

2 3 4

(0

Fig. 1. Ground state energies for H, Hi and HeH*+ enclosed within penetrable spheroidal boxes as a function of box size &,. The nucleus are clamped and R = 1 a.u. The continuous lines represent the variational calculations of this work. The circles are the exact results for VO = 00 reported in Ref. 12.

The energy of the ground state of H, Hz and HeH*+ for Vo = 0, 2, 8, 20 and 00 Ry is dis- played in Fig. 1 as a function of the size of the enclosing box &. As can be noted, good agreement is found with the exact results for the case of V, = 00. When V0 has a finite value, the calculations show the correct qualitative behavior, similar to that found by Ley-koo and Rubinstein [15] for hydrogen within penetrable spherical boxes.

In order to gain some confidence in the value of the ansatz wavefunctions, we have calculated the polarizability and pressure for Hl. In the Kirkwood approximation [16] the parallel and perpendicular components of the polarizability are given by

(“11 = CX, - - 4(3)2 (10)

(Y1 = CX’, = OYY = 4(X2)2 (11)

Finally, the pressure on the system due to the confinement can be calculated through the relationship

dE P'-rv (12)

where V is the volume of the spheroidal box and E is the total ground state energy of the system calculated at the equilibrium bond length. The results obtained for these properties are displayed in Fig. 2 for V0 = 0 and 00 Ry as a function of .$-,. As can be seen, good agreement is found when our results are compared for V0 = co with the calculations reported by LeSar and Herschbach [5,6], who used a five-term James-Coolidge variational wavefunction. For finite values of V, the correct qualitative behavior is obtained [17]. As the reader must be aware, the variational calculations in this work reproduce closely the values of (~11, crl and & reported by LeSar and Herschbach for high pressures. The dis- crepancy observed in the range of moderate to low pressures can be attributed to the ansatz wave- function, which seems not flexible enough in this region.

284 J.L. Marin and G. Murioz/J. Mol. Struct. (Theochem) 287 (1993) 281-285

a, /a.u

2

1

0

(4

I I I

v o=o

6.0 7.5 log ,. b/bar)

WI&p/bar) I I I

300

V/(a.u.)3

all /a.u.

6

0

(b) 6 a

Re / bohr

2.4

1.6

0.8

(4

I

4

I

6

1

a

b&/bar)

Fig. 2. Physical properties of H: enclosed within penetrable spheroidal boxes as a function of box size &,. The continuous lines represent the variational calculations of this work. The circles are the results of Refs. 5 and 6, while the squares those of Ref. 17.

Final remarks

The results presented in this work show that the direct variational method is a suitable tool to study quantum confined systems by either impenetrable or

penetrable boundaries. This approach is clearly economical and physically appealing. Work is in pro- gress in order to consider models of real systems such as electron traps in ionic solids within the variational method and the results will be published elsewhere.

J.L. Marin and G. Mufioz/J. Mol. Strut. (Theo&m) 287 (1993) 281-285 285

Acknowledgments

The authors are greatly indebted to Professors S.A. Cruz and E. Castafio for their valuable comments and suggestions while reviewing the manuscript.

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