Multiperturbation Approach to Potential Energy Surfaces for Polyatomic Molecules

Multiperturbation Approach to Potential Energy Surfaces for Polyatomic Molecules
Multiperturbation approach to potential energy surfaces for polyatomic
molecules
Donald H. Galvan, Moh’d AbuJafar, and Frank C. Sanders
Citation: J. Chem. Phys. 102, 4919 (1995); doi: 10.1063/1.469540
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Multiperturbation approach to potential energy surfaces
for polyatomic molecules
Donald H. Galvan
Instituto de Fisica de UNAM, Laboratorio de Ensenada, Apto. Postal 2681, Ensenada,
Baja California 22800, Mexico
Moh’d Abu-Jafara) and Frank C. Sanders
Department of Physics and Molecular Science Program, Southern Illinois University at Carbondale,
Carbondale, Illinois 62901-4401
~Received 22 August 1994; accepted 12 December 1994!
In Z-dependent perturbation theory, the lowest-order wave functions for a polyatomic molecule are
not only independent of the nuclear charges, but also of the total number of nuclear centers and
electrons in the molecule. The complexity of the problem is then determined by the highest order
retained in the calculation. Choosing the simplest possible unperturbed Hamiltonian, we describe an
n-electron, m-center polyatomic molecule as n ‘‘hydrogenic’’ electrons on a single center perturbed
by electron–electron and electron–nucleus Coulomb interactions. With this H 0 , the first-order wave
function for any polyatomic molecule will be a sum of products of hydrogenic orbitals with either
two-electron, one-center or one-electron, two-center first-order wave functions. These first-order
wave functions are obtained from calculations on He-like and H1
2 -like systems. Similarly, the
nth-order wave function decouples so that the most complex terms are just the nth-order wave
functions of all the p-electron, q-center subsystems ( p1q5n12) contained in the molecule. We
illustrate applications of this method with some results, complete through third order in the energy,
for H1
3 -like molecules. These are compared with accurate variational results available in the
literature. We conclude that, through this order, this perturbation approach is capable of yielding
results comparable in accuracy to variational calculations of moderate complexity. The ease and
efficiency with which such results can be obtained suggests that this method would be useful for
generating detailed potential energy surfaces for polyatomic molecules. © 1995 American Institute
of Physics.
I. INTRODUCTION
Z-dependent perturbation theory ~ZDPT! has long been a
powerful computational tool for atomic systems. Little comparable work has been done, however, in molecular systems.
Examples include the work of Goodisman1 and of Matcha
and Byers Brown2 ~on diatomic systems! based on an unperturbed Hamiltonian taken as the sum of one-electron, diatomic ~H1
2 -like! Hamiltonians, and the work of Dvorácek
and Horák3 ~on the hydrogen molecule!, of Chisholm and
Lodge,4 ~on two-electron diatomic systems!, and of Montgomery, Bruner, and Knight5 ~on ten-electron hydrides!, all
of which utilize single-center, hydrogenic Hamiltonians to
describe the unperturbed system. These approaches illustrate
one characteristic of the application of ZDPT to molecules; it
is possible to construct a variety of unperturbed Hamiltonians and still retain the essential features of the theory.
The advantages of ZDPT and a multiperturbation approach for atomic systems are discussed in Sanders.6 The
present paper extends this earlier work to polyatomic systems. One of the advantages of ZDPT is its ability to provide
results for an entire isoelectronic sequence from a single calculation. While this is less significant for polyatomic systems, where few molecular isoelectronic sequences have
more than one or two physically interesting examples, the
a!
Present address: Department of Physics, An Najah University, P.O. Box 7,
Nablus, West Bank via Israel.
J. Chem. Phys. 102 (12), 22 March 1995
method retains some important advantages for molecular
systems. As usual, the inverse nuclear charge appears as a
natural perturbation parameter of the method. In addition, the
ratio of the nuclear charges also appears as a natural choice
of expansion parameter in a multiple perturbation theory.
These ensure a rapid convergence of the multiperturbation
series for molecules possessing at least one heavy atom. Of
greater importance, with these perturbation parameters the
individual multiperturbation wave functions and energy coefficients are independent of the nuclear charges as well as of
the total number of electrons and the overall electronic configuration of the system. Consequently, results obtained for
small systems can be transferred without modification to
larger systems which contain the electron configuration of
the smaller system as a subconfiguration. It is these characteristics of the method that suggest that ZDPT can be an
accurate and efficient method for studying potential energy
surfaces of polyatomic systems.
Chisholm and Lodge4 studied the ground state of twoelectron diatomic systems ~specifically H2 and HeH1!
through second-order in the energy. The present paper extends these calculations to third order in the energy. These
12
results are then incorporated into a study of the H1
3 , HeH2
sequence as the simplest prototype of a polyatomic molecule.
As in Refs. 3–5, we place all the unperturbed ~hydrogenic!
orbitals on the same nuclear charge. Of the various possible
choices of a zero-order Hamiltonian, this produces the simplest form for the higher-order perturbation coefficients. It
0021-9606/95/102(12)/4919/12/$6.00
© 1995 American Institute of Physics
4919
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4920
Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
also means that contributions of the interelectron interaction
can be obtained directly from ZDPT calculations for atoms.
The complete, first-order wave function of any molecule can
then be constructed from the first-order, atomic ~He-like! pair
functions of all two-electron configurations present in the
zero-order wave function, together with the first-order, oneelectron diatomic ~H1
2 -like! wave functions of all the orbitals
in the zero-order wave function. Similarly, the second-order
correction to the energy consists entirely of one-center threeelectron, two-center two-electron, and three-center oneelectron contributions, no matter how complex the molecule.
Continuing to higher-order, the maximum degree of complexity of the calculation increases in a predictable manner,
with each additional order of the calculation introducing either an additional electron or an additional center to the expansion coefficients.
Despite its simplicity, this zero-order Hamiltonian is
clearly not the best choice for many polyatomic molecules,
particularly homonuclear molecules. Nevertheless, it will
serve here to illustrate the general structure of the multiperturbation expansion and the general characteristics of the
method. A more natural choice of zero-order Hamiltonian
would distribute the electrons for the system among the
nuclear centers. This would significantly improve the initial,
unperturbed electron density and hence also improve the
convergence of the perturbation series, particularly as one
approaches the separated-atom limit. Of course, this would
also increase the complexity of the calculation at each order
of the perturbation by precisely the number of additional
centers that have been introduced into the zero-order Hamiltonian. Such generalizations of the method are straightforward and will be examined in more detail in a later paper.
We describe an unperturbed N-electron, M -center polyatomic molecule as N ‘‘hydrogenic’’ electrons on a single
center of charge Z A . In charge-scaled atomic units,7 this
Hamiltonian is written as H5H 0 1H 1 , where
H 05
(
i51
N
H 15
S
S
2
H i1j 5
1
,
rij
D
1
1
D2
,
2 i r iA
N
( (
i51
G 0 c 0 50,
~3!
G 0 c i1j 1G i1j c 0 50,
~4!
G 0 c i1a 1G i1a c 0 50,
~5!
G 0 c i2j 1G i1j c i1j 2 e i2j c 0 50,
~6!
G 0 c i2a 1G i1a c i1a 2 e i2a c 0 50,
~7!
j, jk
j, jk
G 0 c i1,1
1G i1j c 1jk 1G 1jk c i1j 2 e i1,1
c 0 50,
~8!
j,i a
G 0 c i1,1
1G i1j c i1a 1G i1a c i1j 2 e i11j,i a c 0 50,
~9!
a ,i b
a ,i b
G 0 c i1,1
1G i1a c i1b 1G i1b c i1a 2 e i1,1
c 0 50,
~10!
G a1 5H a1 2 e a1 .
where G 0 5H 0 2 e 0 , and
Equations ~4! and ~6!
reduce to two-electron, one-center equations, while Eqs. ~5!
and ~7! simplify to one-electron, two-center equations. Equations ~8!, ~9!, and ~10! are three-electron one-center, twoelectron two-center, and one-electron three-center equations,
respectively. From these perturbation differential equations
one can obtain all corrections to the energy through fifth
order.8 Expressions for the third-order energy coefficients are
presented below. In these expressions, a, b, and c represent
any one of the perturbations i j or i a , with the restriction that
all perturbations appearing in a coefficient must be different:
e a1 5 ^ c 0 u H a1 u c 0 & ,
e a2 5 ^ c a1 u G a1 u c 0 & ,
II. THEORY
N
any case, it is clear that the convergence of portions of the
multiperturbation series will be slower than might be anticipated based on the size of Z A alone.
Treating each term in Eq. ~2! as a separate perturbation,
we obtain the multiperturbation differential equations.6
Through second order, these are
~1!
M
l i j H i1j 1
j.i
H i1a 52
(
a 5B
1
,
r ia
l i a H i1a
D
a
b
e a,b
1,1 52 ^ c 1 u G 1 u c 0 & ,
e a3 5 ^ c a1 u G a1 u c a1 & 22 e a2 ^ c a1 u c 0 & ,
~11!
a
a b
a
b a
a
b
e a,b
2,1 52 ^ c 1 u G 1 u c 1 & 1 ^ c 1 u G 1 u c 1 & 22 e 2 ^ c 1 u c 0 &
a
22 e a,b
1,1 ^ c 1 u c 0 & ,
a
b c
b
a c
a
c
b
e a,b,c
1,1,1 52 ~ ^ c 1 u G 1 u c 1 & 1 ^ c 1 u G 1 u c 1 & 1 ^ c 1 u G 1 u c 1 &
c
a,c
b
b,c
a
2 e a,b
1,1 ^ c 1 u c 0 & 2 e 1,1 ^ c 1 u c 0 & 2 e 1,1 ^ c 1 u c 0 & ! .
,
~2!
21
where l i j 5Z 21
A and l i a 5Z a Z A ; the Z a being the charges
of the other atoms in the molecule. The perturbation expansion coefficients are then independent of the nuclear charges
and completely transferable from one system to another. This
choice of zero-order Hamiltonian also reduces to a minimum
the number of nuclear centers that can appear at any particular order. Note that the ratio of the nuclear charges appears in
these perturbation expansions in addition to the usual inverse
of the nuclear charge. Obviously, Z A should be chosen as the
largest of the nuclear charges in the molecule, if possible. In
Extensions to higher order are straightforward, and examples
can be found in Ref. 6.
It can be shown that all ‘‘unlinked’’ wave functions decouple into simple products of their ‘‘linked’’ components,
e.g.,
12 34
c 12,34
1,1 5 c 1 c 1 ,
a ,2 a
c 11,1
5 c 11 a c 21 a ,
~12!
4a
a
c 12,23,4
5 c 12,23
1,1,1
1,1 c 1 .
From this it follows that all such ‘‘unlinked’’ energy coeffia
cients disappear, e.g., e12,3
1,1 50. These results ensure that at
nth order the wave functions and energy coefficients cannot
involve more than p electrons and q centers, where p1q5n
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Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
12. This in turn means that an nth order perturbation correction can involve no more than n11 coupled electrons
while the greatest number of nuclear centers involved in nth
order is also n11.
A. Charge-scaling in molecules
In contrast to the corresponding expressions for atoms,
the multiperturbation expansions for a molecule have a much
more complex behavior with respect to the nuclear charges.
The multiperturbation expansion for the electronic energy of
a heteronuclear diatomic molecule serves to illustrate this
behavior. In charge-scaled atomic units, this is
E5 e 0 1
(
S
l nB e nM 1Z 21
e A1 1
A
n51
S
1Z 22
e A2 1
A
1••• ,
l B 5Z B Z 21
A .
(
n51
(
l nB e A,M
1,n
n51
D S
l nB e A,M
1Z 23
e A3 1
2,n
A
D
(
n51
l nB e A,M
3,n
D
~13!
where
A comparable expression for the energy
of a polyatomic molecule requires a simple and obvious extension of this form and introduces no new behavior with
respect to charge-scaling. The notation in Eq. ~13! has been
simplified so that each ‘‘molecular’’ coefficient, enM , each
‘‘atomic’’ coefficient, eAn , and all ‘‘mixed’’ coefficients have
implicit in them sums over all N electrons of the molecule.
The first two terms in this expression represent the sum of
the energies of all states of those one-electron diatomic molecules which are subsystems of the molecule, while the sum
over all the first ~‘‘atomic’’! coefficients in each of the bracketed terms represents the energy of the atom which is a subsystem of the diatomic molecule.
Since l B >Z 21
A , the convergence of those parts of the
series involving lB can be slower than that of the purely
‘‘atomic’’ contributions. An obvious exception to this occurs
for the hydrides, where all the multiperturbation expansion
parameters are identically Z 21
A , and the multiperturbation expansion simply becomes a formal device for identifying the
contributions from subsystems of the molecule. Perturbation
energies and other properties of hydrides will thus have
simple charge-scaling behavior. At the other extreme are the
homonuclear molecules, for which all lB 51. Here, the results obtained via perturbation theory can be seriously affected by a premature truncation of the perturbation series.
Note in particular that the entire portion of the energy arising
from purely one-electron diatomic energy coefficients is of
the same order ~in terms of the nuclear charges! as the zeroorder term. Similarly, terms of a particular order in Z 21
A will
contain contributions from multiperturbation coefficients of
much higher nominal order. Hence, for homonuclear molecules, truncating the perturbation sum at some order can
yield a relative error which actually increases as Z A increases. The convergence of the energy can be improved
considerably, however, by including the higher-order coefficients of one-electron diatomics. This is effectively what was
done in the work of Joulakian9 on isoelectronic homonuclear
diatomics.
4921
Finally, it must be kept in mind that all lengths, including internuclear separations, have been scaled by Z A . Hence,
energies calculated at a particular scaled internuclear distance will actually correspond to different internuclear separations for different members of an isoelectronic molecular
sequence according to the relation R~bohr!5Z 21
A 3R~scaled
a.u.!.
B. Degeneracy
In degenerate multiple perturbation theory, the multiperturbation expansion becomes algebraically more complex.8
Nevertheless, the multiperturbation expansion coefficients
can still be constructed from the corresponding coefficients
of smaller subsystems of the molecule. Chisholm and
Lodge10 have described one approach to this problem, demonstrating how these perturbation coefficients can be constructed.
For the ground state of two-electron molecules, the zeroorder wave function is a product of hydrogenic 1s orbitals
and is not degenerate. Thus, in what follows, this system will
serve to illustrate the structure of the method most clearly
and simply. The single-center, zero-order wave functions for
excited states of these systems and for the ground state of
three-electron molecules will be degenerate, however. To illustrate the application of the method to these more complex,
often degenerate systems, a brief discussion of the multiperturbation expressions for H3-like molecules is presented in
the Appendix.
III. APPLICATION TO H1
3 -LIKE MOLECULES
For two-electron, three-center molecules, the complete
perturbation is
H 15
S
F
D S
1
1
1 1
1
1
2Z B
1
2Z c
1
Z A r 12
r 1B r 2B
r 1C r 2C
DG
.
~14!
For Z C 50, this becomes the perturbation for H2-like molecules. Hence, in what follows, all expressions are written
for H1
3 -like systems, expressions for two-electron diatomics
being obtained by simply setting Z C equal to zero. The complete, first-order energy coefficient for the ground state of the
molecule is
1
1s s
s
e 1 5 e 11 S 12Z B e 1s
1 ~ R B ! 12Z C e 1 ~ R C ! .
~15!
11S
Here e 1 5 85 is the first-order correction to the ground state
energy of a two-electron atom ~and corresponds to an ei1j !
while
s
e 1s
1 ~ R ! 52
S D
1
1
1 11 e 22R
R
R
~16!
is the first-order correction to the ground state energy of a
one-electron diatomic molecule with internuclear distance R
~and corresponds to an ei1a !. R B and R C are the internuclear
distances between the charge Z A and the two perturbing
charges, Z B and Z C . The angle subtended by these two
charges is denoted by Q.
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4922
Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
Each term in the perturbation, Eq. ~3!, yields a corresponding term in the complete first-order wave function
1
2 1s s
s
e 2 5 e 12 S 12Z 2B e 1s
2 ~ R B ! 12Z C e 2 ~ R C !
1
s ,1s s
12Z B Z C e 1s
~ R B ,R C ,Q ! ,
1,1
1s s
s
11s ~ r1 ! c 1s
1 ~ R B ,r2 !# 1Z C @ c 1 ~ R C ,r1 ! 1s ~ r2 !
s
11s ~ r1 ! c 1s
1 ~ R C ,r2 !# .
~17!
1
c 11 S can be obtained accurately from variational perturbation
s
can be expressed in closed form in
calculations11 while c1s
1
confocal elliptic coordinates,12 but is here also obtained from
a variational perturbation calculation.
With this wave function, the second- and third-order energy coefficients can be calculated. The second-order energy
is given by
1
1
1Z B e 11,1S,1s s ~ R B ! 1Z C e 11,1S,1s s ~ R C !
1
s
c 1 5 c 11 S ~ r1 ,r2 ! 1Z B @ c 1s
1 ~ R B ,r1 ! 1s ~ r2 !
1
~18!
1
s
where e 12 S 5 20.157 666 4,11 e1s
is known exactly,12
2
1 1 S,1s s
and e 1,1
is given in Ref. 4. Note that the dependence of
the electronic energy on the angle Q makes its first appearance at this order through the three-center term,
K
U
s ,1s s
s
e 1s
52 c 1s
1,1
1 ~ R B ,r1 ! 2
L
U
1
1s ~ r1 ! .
r 1C
~19!
The third-order energy is given by
1
1
1
3 1s s
1 S,1s s
s
e 3 5 e 13 S 12Z 3B e 1s
~ R B ! 1Z C e 12,1S,1s s ~ R C ! 1Z 2B e 11,2S,1s s ~ R B ! 1Z 2C e 11,2S,1s s ~ R C !
3 ~ R B ! 12Z C e 3 ~ R C ! 1Z B e 2,1
1
s ,1s s
s ,1s s
S,1s s ,1s s
12Z 2B Z C e 1s
~ R B ,R C ,Q ! 12Z B Z 2C e 1s
~ R B ,R C ,Q ! 1Z B Z C e 11,1,1
~ R B ,R C ,Q ! .
2,1
1,2
All of the singly-subscripted coefficients above are either
known exactly13 or are known to high precision from variational perturbation calculations.11 The remaining multiperturbation coefficients can be computed, via Eqs. ~11!, from the
appropriate components of the first-order wave function.
The total energy through nth order, in atomic units, is
given by
~20!
c 11 S ~ r1 ,r2 ! 5 ~ 11 P 12 ! ( c nml r n1 r m2 P l ~ cos u 12 !
1
nml
[email protected] 2 b l ~ r 1 1r 2 !# ,
n
s
c 1s
1 ~ R,ri ! 5 ( c nl r i P l ~ cos u i ! exp~ 2 b 8
l ri!.
~23!
~24!
nl
n
E n ~ R B ,R C ,Q ! 5Z 2A
(
p
Z2
A e p1
p50
1
Z AZ B Z AZ C
1
RB
RC
Z BZ C
.
R BC
~21!
The total wave function, truncated through first order, can
also be used to obtain a variational bound on the energy
E v 5Z 2A e 0 1Z A e 1 1
1
Z BZ C
.
R BC
e 2 1 e 3 /Z A
11Z 22
A ^ c 1u c 1&
1
Z AZ B Z AZ C
1
RB
RC
~22!
Results for both the third-order perturbation sums and this
variational energy for H1
3 -like systems are presented in the
tables, where they are compared with accurate, variationally
obtained energies.
IV. METHOD
The first-order wave functions required by the method
have been obtained variationally. To simplify the calculation
of the multicenter integrals that appear in the energy expansion coefficients, we have used single-center basis sets for all
wave functions. Hence,
For the ‘‘atomic’’ wave function, Eq. ~23!, all terms with
l<16 and n1m12l<20 were included for a total of 501
terms. For the ‘‘molecular’’ wave function, Eq. ~24!, all 221
terms with l<16 and l1n<20 were utilized. For both wave
functions, the nonlinear parameters, bl , of each partial wave
were separately optimized. Once obtained, the optimized
‘‘atomic’’ wave function is stored. The nonlinear parameters
for the ‘‘molecular’’ wave function, however, must be optimized for each value of the internuclear distance, R. For the
sake of efficiency, these parameters were obtained at intervals of 0.10 bohr ~and at smaller intervals near the equilibrium distances!. These were then used to interpolate for the
parameters at other values of R. Since the calculation of
these one-electron, ‘‘molecular’’ wave functions is extremely
rapid, it costs little in efficiency to simply recalculate them as
needed. Hence, none of these ‘‘molecular’’ wave functions
were stored and only the interpolation table for the nonlinear
parameters was saved.
With these wave functions in hand, it only remains to
calculate the energy expansion coefficients for each particular choice of R B , R C , and Q. These can be calculated very
efficiently; for each such point, the entire calculation of the
energy through third order for all values of Z of interest
consumes about 20 s of cpu time on an IBM 9021 vector
processor with 2 cpu’s. Hence, to calculate all the data actually presented in the tables to follow requires less than 20
min of cpu time. Even greater efficiency is possible in cal-
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Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
4923
TABLE I. Some perturbation energy coefficients ~in a.u.!; a comparison of
the present results with exact values.a
R ~bohr!
s b
2 e 1s
2
s c
2 e 1s
3
e 11,1S,1s s d
s ,1s s
e 1s
1,1
0.40
0.386 56
0.386 57
0.312 05
0.312 08
0.249 18
0.249 21
0.199 27
0.199 31
0.160 29
0.160 33
0.070 19
0.070 25
0.025 67
0.025 69
0.009 769
0.009 774
20.022 30
20.022 29
20.017 50
20.017 44
20.005 93
20.005 86
0.005 97
0.005 99
0.015 08
0.015 17
0.023 14
0.023 25
0.011 02
0.011 07
0.003 59
0.003 61
0.529 45
0.529 60
0.448 98
0.449 10
0.367 98
0.368 11
0.293 88
0.294 00
0.229 98
0.230 08
0.075 03
0.075 09
0.015 36
0.015 39
0.002 81
0.002 82
20.583 15
20.583 17
20.339 32
20.339 37
20.160 36
20.160 37
20.044 58
20.044 60
0.022 67
0.022 68
0.068 73
0.068 74
0.032 972
0.032 974
0.013 910
0.013 911
0.60
0.80
1.00
1.20
2.00
3.00
4.00
1
e
a
Each first entry below corresponds to the present results.
The second entry in this column corresponds to the results of Ref. 12.
c
The second entry in this column corresponds to the results of Ref. 13.
d
The second entry in this column corresponds to the results of Ref. 4~b!.
e
These results are for Q5180°, the second entry in this column corresponding to the results of Ref. 14.
b
culating the potential energy surfaces presented in the figures. For each value of R B , R C , the energies for all Q of
interest are calculated together. Since all coefficients that do
not depend on Q are not recalculated, less than 10 s of cpu
time is required for each point. Additional savings in computational time are possible whenever R B 5R C so that many
of the multiperturbation coefficients are identical.
s
s
and e1s
In Table I we compare exact values12,13 of e1s
2
3
1s s
with those obtained with our approximate c1 and find that
the approximate coefficients are accurate to at least four
decimal places over the entire range of internuclear distances
considered here. The simple wave function utilized here for
1
1
1
c 11 S yields values for e 12 S and e 13 S which agree with accurate values of these coefficients to three decimal places. In
Table I we also compare calculated and exact values4 of
1
e 11,1S,1s s . Here the results obtained with the ‘‘atomic’’ firstorder wave function are accurate to four decimal places, except for R<1, where they are in error by roughly one unit in
the fourth decimal place. ~These are obtained with greater
s
accuracy if calculated with c1s
1 .! Comparing exact values of
1s s ,1s s
e1,1
~Ref. 14! with those calculated with the approximate
first-order ‘‘molecular’’ wave function shows that the latter
agree with the exact values to four or five decimal places.
Not surprisingly, the approximate one-electron ‘‘molecular’’
first-order wave functions are more accurate than the twoelectron ‘‘atomic’’ wave functions, but between them, they
appear capable of calculating the ‘‘mixed’’ perturbation energy coefficients to roughly four decimal places for the range
of R considered here. Note that, in calculating the total energies, only exact values of these coefficients have been used
where they are available. Hence, the total energies presented
in the tables cannot be expected to agree with accurate variational energies to more than four decimal places. Conversely,
if the present calculations agree with accurate variational en-
FIG. 1. Energy for H2 ; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s! variational
results of Ref. 15.
ergies by less than about four decimal places, the discrepancy can be attributed to the truncation of the perturbation
series.
V. RESULTS AND DISCUSSION
Although the main focus of this work is the calculation
of polyatomic molecular energies, a brief examination of results for diatomic molecules will illustrate the behavior of
the multiperturbation series in the present method.
A. Diatomic molecules: H2
A comparison of the present perturbation results with
accurate variational calculations15 for the ground state of the
H2 molecule is presented in Fig. 1. In this figure, the convergence of the perturbation results can be gauged by examining
the behavior of successive truncated perturbation sums, the
E n of Eq. ~21!, for the total energy of the molecule. These
results indicate that the variational energy, E v , obtained with
the first-order wave function is consistently less accurate
than the corresponding truncated third-order energy sum, E 3 ,
obtained with the same wave function. This is not unexpected, as variational expressions based on perturbation
wave functions such as that of Eq. ~22! seem to consistently
yield poorer results than the corresponding truncated perturbation series. In fact, in this instance at least, E v is less
accurate than E 2 . It is also interesting to note that E 2 is quite
similar to the Hartree–Fock result16 for this molecule.
The present calculations have been carried out in sufficient detail to verify that all the E n and E v have a minimum
in the potential energy close to the equilibrium internuclear
distance for this molecule. E 3 yields the best result at
R51.36 bohr, within 0.04 bohr of the correct value of 1.40
bohr. Figure 2 displays differences between the ‘‘exact’’
variational results and the perturbation results,
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4924
Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
FIG. 2. Truncation error, Dn , for H2 ; ~n! D1 ; ~L! D2 ; ~h! D3 ; ~!! E v .
D n 5E n 2E exact ,
showing that the perturbation results are most accurate in the
vicinity of the energy minimum. For example, the smallest
error in E 3 occurs at R51.2 bohr where it is 0.006 a.u., or
0.5%. For larger R, all the perturbation results become increasingly less accurate, so that by R53 the error is approximately 0.1 a.u. By R55 the error for E 2 , E 3 , and E v has
grown to '0.25 a.u., a relative error of just over 25%. The
present method fails to give accurate results for large R for
this molecule since the zero-order wave function tends to an
incorrect separated-atom limit ~H21H1!. H2 is, of course, a
particularly difficult case for the Z 21 expansion. Nevertheless, the convergence of the perturbation expansion for this
particular molecule is quite satisfactory. For this system, e3 is
negative for the entire range of R presented in the table except for the smallest R and it provides a considerable improvement over the second-order results of Chisholm and
Lodge,4 reducing the error in the energy by almost an order
of magnitude. In fact, the present results show that the relative error for E 3 remains under 1% for 1&R&2 bohr.
In general, use of similar zero-order wave functions in
homonuclear molecules should produce perturbation series
which converge most rapidly for intermediate values of the
internuclear distance, but become suspect for larger values
of R.
FIG. 3. Energy for HeH1; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s! variational
results of Ref. 17.
at the equilibrium distance. Indeed, E 3 does not yield quite as
good an equilibrium distance as either E 2 or E v ; R51.41 as
compared to R51.50 and 1.43, respectively. All however lie
within 0.05 bohr of the correct value of R51.463 ~Ref. 17!.
In contrast to the other systems studied here, E v is consistently better than both E 2 and the Hartree–Fock energy18 for
R&2. Again, as in the case of H2 , E 2 and the Hartree–Fock
energy are quite similar for R&2, beyond which point E 2 is
a significant improvement over E v . Since the perturbation
series for this molecule converges more rapidly, the improvement that E 3 provides over E 2 is not as large here as in the
case of H2 . Nevertheless, E 3 generally reduces the error relative to accurate variational results17 to less than half what it
was for E 2 . Over the entire range of R, the relative error in
E 3 is less than 1% for this molecule. Indeed, it is less than
0.5% for 1.0*R*2.5.
B. Diatomic molecules: HeH1
Figures 3 and 4 present results for HeH1 in a form similar to those for H2 . For this molecular ion, all perturbation
parameters are equal to 21 and a more rapid convergence of
the perturbation series can be expected. Moreover, unlike the
case of H2 , the zero-order wave function does represent the
separated-atom limit correctly. Thus, for large R, the perturbation results are very well converged except for E 1 , where
1
the discrepancy is primarily due to the absence of e 12 S . For
this system, we find that E 2 and E 3 give slightly better results
at both larger and smaller internuclear distances rather than
FIG. 4. Truncation error, Dn , for HeH1; ~n! D1 ; ~L! D2 ; ~h! D3 ; ~!! E v .
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Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
4925
TABLE II. Total energies ~in a.u.! for the H1
3 molecule; the first-order variational energy, E v , and truncated energy sums, E n , compared with variational
energies.
D 3h Symmetry
R ~bohr!
1.137
1.250
1.277
1.414
1.549
1.644
1.650
1.656
1.819
1.900
2.050
2.094
2.200
2.235
2.750
3.052
3.350
4.500
5.500
7.000
R ~bohr!
1.130
1.257
1.431
1.435
1.438
1.452
1.462
1.507
1.509
1.537
1.571
1.609
1.619
1.695
1.743
1.756
1.756
1.794
1.848
1.894
1.962
1.995
2.085
2.154
2.277
2.419
3.092
4.067
5.054
6.044
8.031
10.025
a
Ev
E1
E2
E3
E~variational!a
20.913
21.015
21.033
21.105
21.145
21.159
21.160
21.161
21.166
21.163
21.150
21.145
21.130
21.125
21.032
20.975
20.923
20.776
20.706
20.649
20.481
20.584
20.603
20.678
20.724
20.743
20.744
20.745
20.762
20.765
20.764
20.763
20.758
20.756
20.716
20.691
20.667
20.597
20.557
20.518
21.265
21.282
21.283
21.280
21.265
21.249
21.248
21.247
21.215
21.197
21.163
21.153
21.128
21.120
21.009
20.954
20.907
20.787
20.730
20.681
21.230
21.304
21.317
21.365
21.387
21.391
21.391
21.391
21.381
21.370
21.343
21.334
21.310
21.302
21.166
21.088
21.018
20.831
20.746
20.682
21.233 78
21.286 40
21.295 48
21.327 44
21.341 24
21.343 79
21.343 83
21.343 78
21.338 33
21.331 39b
21.319 13
21.314 56
21.301 51b
21.298 63
21.234 12b
21.198 43b
21.166 32b
21.076 51b
21.036 26b
21.011 21b
Q
124.141
117.047
114.447
109.417
132.515
99.513
95.863
82.960
157.418
76.835
70.958
65.344
103.498
54.929
50.129
20.043
99.957
45.594
41.316
23.069
33.488
31.510
26.551
68.199
29.448
31.275
75.964
79.588
81.637
83.063
84.928
85.939
Ev
21.041
21.117
21.174
21.174
21.180
21.175
21.176
21.179
21.195
21.179
21.176
21.170
21.194
21.165
21.166
20.353
21.193
21.164
21.156
20.872
21.122
21.107
21.050
21.175
21.130
21.153
21.138
21.114
21.104
21.102
21.102
21.101
C 2 v Symmetry
E1
20.857
20.918
20.969
20.957
21.004
20.931
20.920
20.875
21.037
20.847
20.817
20.783
20.963
20.764
20.781
0.310
20.955
20.795
20.805
20.606
20.807
20.802
20.771
20.828
20.840
20.859
20.870
20.877
20.878
20.878
20.880
20.880
E2
21.307
21.313
21.298
21.305
21.270
21.315
21.317
21.316
21.235
21.309
21.295
21.275
21.283
21.262
21.273
20.320
21.261
21.281
21.284
21.150
21.274
21.266
21.230
21.229
21.244
21.234
21.175
21.143
21.129
21.123
21.122
21.121
E3
21.131
21.205
21.254
21.262
21.239
21.282
21.291
21.328
21.237
21.348
21.365
21.380
21.285
21.385
21.375
20.784
21.287
21.361
21.341
21.029
21.283
21.263
21.193
21.343
21.265
21.285
21.243
21.195
21.177
21.170
21.167
21.166
E~variational!a
21.238 38
21.278 81
21.301 07
21.305 22
21.289 43
21.314 26
21.317 79
21.330 28
21.281 33
21.335 63
21.339 92
21.342 76
21.309 96
21.342 62
21.338 81
20.889 35
21.304 81
21.331 83
21.321 02
21.086 63
21.284 03
21.269 23
21.215 80
21.316 90b
21.267 52
21.280 59
21.245 92b
21.201 25b
21.182 25b
21.176 28b
21.174 04b
21.173 45b
Unless otherwise noted, these are all from Frye et al., Ref. 19.
Talbi and Saxon, Ref. 20.
b
C. Triatomic molecules: H1
3
As in the case of H2 , this molecule presents a severe test
for the method. In addition to slow convergence, the calculated ground state energies also have the same difficulties at
the separated-atom limit noted for H2 . Table II presents results for H1
3 in D 3h ~equilateral triangle! and C 2 v ~isosceles
triangle! symmetry. In addition, Figs. 5, 6, and 7 present
potential energy surfaces based on E 1 , E 2 , and E 3 for C 2 v
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4926
Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
FIG. 5. E 1 potential energy surface for H1
3 ; contours are 0.05 a.u. apart
starting at 21.19 a.u.
FIG. 7. E 3 potential energy surface for H1
3 ; contours are 0.05 a.u. apart
starting at 21.39 a.u.
symmetry. In Fig. 5, we see that E 1 does not reproduce the
correct equilibrium separation for D 3h symmetry. Surprisingly, however, it does yield an equilibrium distance of
R51.62 at the minimum of its potential energy surface, close
to the correct value of R51.65, but this occurs for a linear
geometry. Note that the only dependence on Q at this order is
through the nuclear potential, Z B Z C /R BC . Adding the
second-order energy introduces the first Q-dependent contribution to the electronic energy. The overall second-order coefficient is everywhere negative and becomes more so for
small R and Q. This places the minimum of the potential in
Fig. 6 at R51.20, Q590°, shifting the minimum in the right
direction but by too much in R and not enough in Q. The
third-order energy coefficient itself has a minimum very
close to the correct equilibrium distance for the molecule,
but at Q50°. As a consequence, this coefficient produces a
minimum in the potential energy very close to the correct
equilibrium separation but slightly too low in Q. The potential energy surface of Fig. 7 shows a minimum energy at
R51.66 and Q554°. In contrast, the potential produced by
E v and displayed in Fig. 8 shows a long trough with a shallow secondary minimum at its head near Q590° and the
minimum at its base at Q5180°. Of course the energy for
this minimum lies well above those of E 2 and E 3 .
A number of variational calculations have been carried
out to an accuracy of a few thousandths a.u. or better for this
molecule. Notable among these are the calculations of Frye
et al.,19 who obtain very accurate results over a range of
internuclear separations and geometries. Total energies for
larger internuclear distances can be found in Talbi and
Saxon.20 The accuracy of these calculations permit an unambiguous assessment of the convergence of the perturbation
results. A comparison of these results for D 3h and C 2 v sym-
FIG. 6. E 2 potential energy surface for H1
3 ; contours are 0.05 a.u. apart
starting at 21.21 a.u.
FIG. 8. E v potential energy surface for H1
3 ; contours are 0.05 a.u. apart
starting at 21.19 a.u.
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Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
FIG. 9. Truncation error, D3, for H1
3 ; ~n! Q,50°; ~L! 50°,Q,70°; ~s!
70°,Q,90°; ~h! 90°,Q,120°; ~!!Q.120°.
metries is presented in Table II. Figure 9 displays the differences between the variational values and E 3 , grouped according to the value of Q. In the vicinity of the equilibrium
distance, most points fall in the range of 60.05 a.u., a relative error of about 3.5%. Exceptions to this occur for those
points with both the largest and the smallest values of Q; i.e.,
all those with Q.120° and two points belonging to the
group with Q,50° that actually have the smallest values of
Q~'20°!. These all have relative errors ranging from 5% to
10%. At larger internuclear distances, points with Q560°
show an error which increases with R, while points which
correspond to Q'90° have an error which remains fairly
constant at about 0.006 a.u. It is clear that much of this
behavior is determined by the initial charge distribution imposed by the zero-order function. Thus, those points with
R.3 and Q'90°, all of which have geometries with one
internuclear distance held close to the equilibrium separation
for H2 ,20 have errors which are quite close to that of the H2
molecule at equilibrium, the perturbation coefficients involving large internuclear separations contributing little to the
total energy. Conversely, those points at large R corresponding to an equilateral geometry all have errors which increase
with R in a manner similar to the results for H2 at large R.
4927
FIG. 10. Energy for equilateral H1
3 ; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s!
variational results, Ref. 17.
steadily for increasing R, reaching 5% in the vicinity of
R'2.7 and 10% by R'3. However, for 1,R,2.5, the relative error is always less than 3.5%. Both E 3 and E v ~but not
E 2! predict a minimum in the potential energy for this symmetry at the correct internuclear distance to within 60.01
bohr. Surprisingly, E 3 is more negative than the ‘‘exact’’
variational values for 1.25,R,2.5, reaching its greatest discrepancy in the vicinity of the equilibrium position. However, as in the case of H2 , these results show a satisfactory
convergence and a significant improvement in the total energy over the second-order results.
For nonsymmetric geometries of this molecule, comparison to the results of Ref. 19 shows that the truncation error is
generally about 0.02–0.05 a.u. for the internuclear distances
considered there, with the range from largest to smallest being from 0.006 to 0.09 a.u.
D. Equilateral H1
3
For the equilateral geometry, a comparison to H2 is useful throughout the entire range of R. Figures 10 and 11
present the results for equilateral H1
3 in a manner similar to
Figs. 1 and 2 for H2 . Comparison of Figs. 2 and 11 indicates
that, in this case, the errors for E n in H1
3 behave very much
like the corresponding errors in H2 , although these are, in
general, considerably smaller for H2 . For example, for R in
the range 1.2,R,1.8, errors in H1
3 are roughly 2 to 5 times
larger than those found for H2 at similar internuclear distances. Ultimately, and again as in the case of H2 , as R
increases these results converge more slowly and to the
wrong limiting value. Thus, the relative error in E 3 increases
FIG. 11. Truncation error, Dn , for equilateral H1
3 ; ~n! D1 ; ~L! D2 ; ~h! D3 ;
~!! E v .
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4928
Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
TABLE III. Total energies ~in a.u.! for the linear, symmetric HeH11
molecule; the first-order variational energy,
2
E v , and truncated energy sums, E n , compared with the variational results.
R ~bohr!
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Ev
E1
E2
E3
E~variational!a
22.188
22.552
22.653
22.700
22.730
22.753
22.770
22.784
22.795
22.030
22.390
22.497
22.550
22.583
22.607
22.625
22.639
22.650
22.181
22.554
22.660
22.710
22.742
22.765
22.783
22.797
22.808
22.197
22.564
22.664
22.710
22.739
22.762
22.779
22.792
22.803
22.155 91
22.527 73
22.625 87
22.663 36
22.683 70
22.698 66
22.711 56
22.722 94
22.732 74
a
Zetik and Poshusta, Ref. 21.
E. Triatomic molecules: HeH12
2
Comprehensive variational calculations for this unstable
system over a range of internuclear distances can be found in
Zetik and Poshusta21 who examined the potential energy surface for a linear geometry. These results are compared with
the present calculations in Table III and Fig. 12 for the linear,
symmetric molecule. This comparison shows that E 2 , E 3 ,
and E v all lie below this variational calculation by roughly
0.05 a.u. Since E v is itself an upper bound to the exact energy, it is clear that all of these results represent an improvement over the variational result. Note that both E 2 and E 3 lie
lower in energy than E v for R>1, with E 2 being the more
negative for R.2.5. The absolute accuracy of these results is
difficult to judge in the absence of more accurate variational
calculations. Recalling the comparable behavior of the perturbation series for H1
3 and H2 noted earlier, a corresponding
and HeH1 series might be
similarity between the HeH11
2
expected. Indeed, the E n of Table III do suggest a rapid
convergence of the perturbation series for HeH11
2 . In particular, e3 for this system is virtually identical to e3 for HeH1
once R exceeds about 2 bohr, indicating that contributions
from multiperturbation coefficients with two ‘‘molecular’’
FIG. 12. Energy for linear, symmetric HeH11
2 ; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ;
~!! E v ; ~s! variational results of Ref. 21.
perturbations, as appear in polyatomic systems, are quite
small for large R. In fact, the only significant contribution to
1
e3 in this range of R is from e 13 S . In this context, it should
be remembered that internuclear distances are scaled by the
nuclear charge, Z A , so that these energy coefficients are actually calculated with values of R that are twice their nominal value in bohr.
Similar improvements in convergence can be expected
for molecules containing heavier atoms than those appearing
in these two-electron systems. This should be particularly
true for heteronuclear molecules, especially those with a
single heavy ion. For homonuclear molecules, where the
multiperturbation parameter lM 51, the relative error ~in
charge-scaled atomic units! can actually increase with Z
along a particular isoelectronic sequence, while the absolute
error remains roughly constant. In nonscaled atomic units,
M
. This error is easily
the absolute error increases as Z 2A e n11
corrected, however, by simply adding the ‘‘missing’’ portion
of the electronic energy of the appropriate states of the oneelectron diatomic molecule.
VI. SUMMARY
In Z-dependent perturbation theory, the lowest-order
wave functions for a polyatomic molecule are not only independent of the nuclear charges, but are also independent of
the total number of nuclear centers and electrons in the molecule. The complexity of the problem is then determined by
the highest order of the calculation. With the present choice
of H 0 , the first-order wave function for any polyatomic molecule is described completely in terms of two-electron, onecenter ~atomic! and one-electron, two-center ~molecular!
first-order wave functions. These are separately obtained
from calculations on He-like and H1
2 -like systems. At nthorder the wave function for a polyatomic molecule decouples
into a sum over the nth-order wave functions of all
p-electron, q-center subsystems (p1q5n12) that are contained within the molecule of interest.
We have illustrated the application of this method with
some results, complete through third order in the energy, for
H1
3 -like molecules. In applying this method, we have chosen
to describe the system by a zero-order wave function that
minimizes the complexity of the calculations. The perturbation series calculated through third order essentially yield the
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Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
correct equilibrium internuclear distance and geometry, even
in the case of H1
3 . For this molecule, the perturbation series
is best converged in the vicinity of the equilibrium distance.
For larger R, the truncation error remains small if one of the
internuclear distances is held near the equilibrium distance.
However, if all internuclear distances are increased, the truncation error increases as a consequence of the incorrect
separated-atom limit of the zero-order wave function. In the
case of HeH11
2 , the convergence of the perturbation series is
significantly improved, particularly in the vicinity of the
separated-atom limit.
The present calculations can be improved without significant complication of the method by introducing a
screened nuclear charge for the unperturbed Hamiltonian22 or
by repositioning the unperturbed, single-center Hamiltonian
away from the physical charges.3 Calculations incorporating
these improvements and extending this initial study to
higher-order are currently underway.
For all diatomic and polyatomic systems examined here,
the perturbation energy summed through third order generally yields results comparable in accuracy to variational calculations of moderate size and complexity. Given the simplicity and efficiency of the method, these results for simple
systems are very encouraging. Since, through a given order
in the perturbation, the computations are of a fixed degree of
complexity regardless of the size of the molecule, application
to larger systems with higher Z seems promising.
4929
1s s
2l s
e m,l
1 ~ R a ! 52 e 1 ~ R a ! 1 e 1 ~ R a ! .
1,3
Here the e 21 L are the first-order corrections to the singlet
and triplet 2S and 2 P states of the two-electron atom, while
K U U L
s
e 2l
1 ~ R a ! 52 2l ~ r !
1
2l ~ r! ,
ra
s
e 2s
1 ~ R !5
1
1
2
~ 11 43 R1 14 R 2 1 18 R 3 ! e 2R ,
4R 4R
s
e 2p
1 ~ R !5
3
1
1
R 3 4R
2
F
G
3
1
~ 71 114 R1 34 R 2 1 18 R 3 ! e 2R .
3 ~ 11R ! 1
R
4R
The off-diagonal matrix
1Z C v 2s,2p (R C ), with
V5Z B v 2s,2p (R B )
K U U L
F
G
v 2s,2p ~ R ! 52 2s ~ r!
52
element,
1
2 p ~ r!
ra
3
3
~ 11R !
21
4R
4R 2
1 81 ~ 31R1 41 R 2 ! e 2R .
The complete first-order wave function is then given by
APPENDIX: THREE-ELECTRON MOLECULES
c 1 5c s c s1 1c p c 1p 1 gc 80 ,
where
The zero-order wave function for the three-electron,
three-center molecule is degenerate, while a more complex,
permutational symmetry not present in the ground state of
the corresponding two-electron molecule also appears here.
Expressions for the first-order wave function and secondorder energy of this system are obtained here to illustrate
how these affect the multiperturbation expansion. The
doubly-degenerate zero-order wave function for this system
is
c 0 5c s c s0 1c p c 0p ,
where
c l0 52 2 ~ 1/2 ! ~ 12 P 13 ! 1s ~ r1 ! 1s ~ r2 ! 2l ~ r3 !
for l5s or p. The coefficients, c l , are the eigenvectors of the
perturbation matrix formed with these degenerate wave functions. The eigenvalues are just the first-order energies,
e 15
e s1 1 e 1p e s1 2 e 1p
6
2
2
A S
12
2V
e s1 2 e 1p
2
,
appears as an admixture of the other degenerate, zero-order
wave function ~without, however, contributing to the secondorder energy!. Also,
m,l
m,l
c l1 5 c a,l
1 1Z B c 1 ~ R B ! 1Z C c 1 ~ R C ! ,
are the first-order wave functions for the
where the ca,l
1
ground and lowest 2P-state of three-electron atoms. These
can be expressed completely in terms of the first-order wave
functions of two-electron atoms,23
1
2 ~ 1/2 !
c a,l
~ 12 P 13 ! $ c 11 S ~ r1 ,r2 ! 2l ~ r3 ! 12 2 ~ 1/2 !
1 52
1
3
3 @ c 21 L ~ r2 ,r3 ! 1s ~ r1 ! 1 c 21 L ~ r2 ,r3 ! 1s ~ r1 !
3
1 c 21 L ~ r1 ,r3 ! 1s ~ r2 !# % .
The c m,l
1 (R a ) can be constructed entirely from first-order,
H1
2 -like wave functions,
s
2 ~ 1/2 !
c m,l
~ 12 P 13 [email protected] c 1s
1 ~ R a ! 52
1 ~ R a ,r1 ! 1s ~ r2 ! 2l ~ r3 !
where
m,l
m,l
e l1 5 e a,l
1 1Z B e 1 ~ R B ! 1Z C e 1 ~ R C ! ,
with
1
1 S
e a,l
1 5e1 1
and
D
c 08 52c p c s0 1c s c 0p
1 21L 3 23L
e 1 e1 ,
2 1
2
s
1 c 1s
1 ~ R a ,r2 ! 1s ~ r1 ! 2l ~ r3 !
s
1 c 2l
1 ~ R a ,r3 ! 1s ~ r1 ! 1s ~ r2 !# ,
s
are solutions to the degenerate, first-order, perwhere c2l
1
turbation differential equation for the 2l zero-order wave
function,10 e.g.,
J. Chem. Phys., Vol. 102, No. 12, 22 March 1995
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4930
Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules
s
G 0 c 2s
1 ~ R a ,r ! 2
S
D
1
This first-order wave function then yields the second-order
energy coefficient,
1
~ s, p ! s
e a,2
~ R a ! 5 21 @ e 21,1S,2p s ~ R a ! 1 e 21,1P,2s s ~ R a !#
1,1
1
s
1 e 2s
2s ~ r! 2 v 2s,2p ~ R a ! 2p ~ r! 50.
1
ra
3
3
1 23 @ e 21,1S,2p s ~ R a ! 1 e 21,1P,2s s ~ R a !# ,
and
1s 2
e 2 5 e 2 1c 2s e s2 1c 2p e 2p 1c s c p e s,p
2 ,
s ,2 p s
e 2s
~ R B ,R C ,Q ! 522
1,1
2
where e 1s
2 is given by Eq. ~18!, the second-order energy for
the ground state of an H1
3 -like molecule,
2 2l s
2 2l s
a,2l s
e l2 5 e a,l
2 1Z B e 2 ~ R B ! 1Z C e 2 ~ R C ! 1Z B e 1,1 ~ R B !
K
FK
U U L
U U LG
s
c 2s
1 ~ R B ,r !
s
1 c 2p
1 ~ R B ,r !
1
2p ~ r!
rC
1
2s ~ r!
rC
1,3 L,2l s
2
s
One approach to evaluating the e2l
2 , e 1,1
coefficients can be found in Ref. 10.
2l s ,2l s
s
1Z C e a,2l
~ R B ,R C ,Q ! ,
1,1 ~ R C ! 1Z B Z C e 1,1
.
s
, and e2(s,p)
2
and
2 2 ~ s,p ! s
e s,p
~ R B ! 1Z 2C e 22 ~ s,p ! s ~ R C !
2 5Z B e 2
~ s,p ! s
~ s,p ! s
1Z B e a,2
~ R B ! 1Z C e a,2
~RC!
1,1
1,1
s ,2p s
1Z B Z C e 2s
~ R B ,R C ,Q ! .
1,1
2
The multiperturbation coefficients appearing in this
second-order energy coefficient are given by
1
3
1 2 L
3 2 L
l
e a,l
2 5 2e 2 1 2e 2 1 d 2 ,
where d l2 is the three-electron contribution to the secondorder energy of a three-electron atom,23
K
2l s
s
e 2l
2 ~ R a ! 52 c 1 ~ R a ,r !
U U L
1
2l ~ r! ,
ra
1
3
1 2 L,2l s
s
e a,2l
~ R a ! 1 23 e 21,1L,2l s ~ R a !
1,1 ~ R a ! 5 2 e 1,1
with
1,3 L,2l s
e 21,1
K
~ R a ! 522 1/2 c 21
1
U
1,3 L
~ r1 ,r2 !
U
1
r 1a
L
1
1s ~ r1 ! 2l ~ r2 ! 62l ~ r1 ! 1s ~ r2 ! ,
r 2a
K
s ,2l s
s
e 2l
~ R B ,R C ,Q ! 522 c 2l
1,1
1 ~ R B ,r !
K
s
e 22 ~ s,p ! s ~ R a ! 522 c 2s
1 ~ R a ,r !
J. Goodisman, J. Chem. Phys. 47, 1256 ~1967!.
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3
Z. Dvorácek and Z. Horák, J. Chem. Phys. 47, 1211 ~1967!; see also Z. J.
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4
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5
H. E. Montgomery, B. L. Bruner, and R. E Knight, J. Chem. Phys. 56,
1449 ~1972!.
6
F. C. Sanders, Phys. Rev. A 7, 1870 ~1973!.
7
Units of length and energy a 0 /Z and 2RhcZ 2 , respectively.
8
J. Hirschfelder, W. Byers-Brown, and S. T. Epstein, Adv. Quantum Chem.
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9
B. Joulakian, Ind. J. Phys. 64B, 189 ~1990!.
10
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Mol. Phys. 28, 249 ~1974!.
11
C. W. Scherr and R. E. Knight, Rev. Mod. Phys. 35, 1033 ~1963!.
12
A. Dalgarno and N. Lynn, Proc. Phys. Soc. London 70, 223 ~1957!.
13
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14
I. N. Levine, J. Chem. Phys. 41, 2044 ~1964!.
15
W. Kolos, K. Szalewicz, and H. J. Monkhorst, J. Chem. Phys. 84, 3278
~1986!.
16
W. Kolos and C. C. Roothan, Rev. Mod. Phys. 32, 219 ~1960!.
17
W. Kolos and J. M. Peek, Chem. Phys. 12, 381 ~1976!.
18
S. Peyerimhoff, J. Chem. Phys. 43, 998 ~1965!.
19
D. Frye, A. Preiskorn, G. C. Lie, and E. Clementi, J. Chem. Phys. 92,
4948 ~1990!.
20
D. Talbi and R. P. Saxon, J. Chem. Phys. 89, 2235 ~1988!.
21
D. F. Zetik, dissertation, The University of Texas at Austin, 1968; D. F.
Zetik and R. D. Poshusta, J. Chem. Phys. 52, 4920 ~1970!.
22
C. D. H. Chisholm and K. B. Lodge, Mol. Phys. 22, 673 ~1971!.
23
Y. Y. Yung, F. C. Sanders, and R. E. Knight, Phys. Rev. A 48, 74 ~1972!.
1
U U L
1
2l ~ r! ,
rC
U U L
1
2p ~ r! ,
ra
J. Chem. Phys., Vol. 102, No. 12, 22 March 1995
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