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Analytical Evaluation of Three and Four-Center Integrals of r12{}-1 with Slater-Type Orbitals

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ANALYTICAL
EVALUATION
OF
THREE-
AND
FOUR-CENTER
INTEGRALS
OF
r12'-
WITH
SLATER-TYPE
ORBITALS*
BY
HARRIS
J.
SILVERSTONE
DEPARTMENT
OF
CHEMISTRY,
JOHNS
HOPKINS
UNIVERSITY,
BALTIMORE
Communicated
by
Paul
H.
Emmett,
May
11,
1967
Three-
and
four-center
integrals
of
rl2-'
with
Slater-type
orbitals
(STO)
have
long
been
the
"bottlenecks
of
molecular
quantum
mechanics."
Evaluation
of
multicenter
integrals
is
usually
carried
out,
after
some
analytical
manipulations,
by
numerical
integrations,
1-4
except
in
the
special
case
of
linear
triatomic
mole-
cules.5
In
this
communication,
a
technique
is
sketched
for
the
analytical
evaluation
of
arbitrary
three-
and
four-center
integrals,
and
a
particular
three-center
integral
is
evaluated
as
an
example.
The
method
is
beguilingly
simple.
The
first
key
ingredient
is
the
Fourier
trans-
form
convolution
theorem,6-8
which
automatically
reduces
the
six-dimensional
integration
to
a
three-dimensional
one.
The
second
key
ingredient
is
the
expan-
sion2,
3'
9
of
an
STO
on
one
center
about
another
center,
which
reduces
the
Fourier
transform
of
a
two-center
charge
distribution
to
the
sum
of
Fourier
transforms
of
one-center
distributions.
The
technique
is
successful
because
(1)
the
angular
inte-
gration
in
the
convolution
integral
is
over
spherical
harmonics-therefore
easy-
and
(2)
the
radial
integration
can
be
carried
out
by
contour
integration
and
the
residue
theorem.'0
For
an
illustration,
first
consider
the
Fourier
transform
of
the
two-center
product
of
two
is
orbitals,
G(k)
(NbNe)'-lf
dv
exp(ik*r)
ls5(r)
ls,(r
-
i),
(1)
=
(47r)'-f
dv
exp(ik.r
-
xbr
-
PI
r
-
RI).
(2)
Evaluate
(2)
after
expanding
exp(-P4
r
-
(R)
about
the
origin
to
obtain
coD
I
G(k)
=
E
E
Z
7ri
Ye
(,jR.)
YI
(9kk)
1=0
m=-I
X
[
r
]
{5Cz(¢c(R)(4i)-Y(-k)1
(k-
d
)
k'
z
(!
dy
)cc1G12l(Az[(.b-
c
+
ik)&I]
1
-
ik)GR]
-
t21[(¢b
+
Be
+
ik)GR]
+
F2i[(¢b
+
c
-
ik)GR])
+
41(c(R)
(2i)-1k
Ik
dk)
k-'1cl
1
d
rc-131-21
X
(E21[(
b
+
c
-
ik)
(R]
-
E21[(5b
+
Pc
+
ik)G])}.
(3)
In
equations
(2)
and
(3),
Nx
=
2rx3l2,
(kOk,4k)
and
((RO&,,OR)
denote
the
spherical
coordinates
of
k
and
(R,
En
is
an
exponential-type
integral,"
En(x)
=
fl
tn
exp
(-xt)dt,
and
tn
is
the
"entire"
part
of
E.,
tn(x)
=
En(X)
+
(-x)'-1
In
x/(n
-
1)!.
34
CHEMISTRY:
H.
J.
SILVEIGSTONE
The
.4
and
3C,
are
essentially'2
modified
spherical
Bessel
functions:
41(x)
=
x
(x-'d/dx)'x-l
sinh
x;
3C(x)
=
(-x)1(x-'d/dx)1x-l
exp(-x).
Note
particularly
that
the
only
singularities
of
G(k)
are
poles
and
logarithmic
branch
points
at
k
=
-i(rb
+
D.).
Next,
we
use
G(k)
in
the
simplest
example
of
a
three-(noncolinear)-center
inte-
gral,
I
fdvlfdv2
lsa(rl)
lsa(rl)
rl2-'
lsb(r2-
R)
ls,(r2-
R
-
(R).
(4)
Expressed
as
a
convolution
integral,6-8'
1
we
obtain
I
=
1/47r
-12Na2NbNtVfd~kF(k)G(k)k-2
exp(ik
R),
(5)
where
F(k)
-
(4,r)-1I2fdv
exp(ik
r
-
2rar)
=
(4,r)-1/2
2r(ik)-'[(Z
-
ik)-2
(Z
+
ik)
-2]
(see
eq.
(13)
of
ref.
8),
and
where
Z
-
2ra.
Denote
the
spherical
co-
ordinates
of
R
by
(R,OR,4R),
let
cos9
=
R
-.
/(RGI),
and
let
PI
denote
the
Legendre
polynomial
of
order
1.
Carrying
out
the
integration
of
equation
(5)
for
the
case
R
>
6R,
we
obtain
I
=
Na2NbNc
:
(21
+
1)
(-1)
'Pi(cosO)
I
=o
X
RI
(
d)
R-'(-dld~c)¢ct.4(¢ca)I,10
+
Xc1(rcM)I2(')}I
(6)
Ii()(21'+'1!
Z3
1
(1
d\
I
I(¢b
+
r.)-2[1
+
(¢b
+
r-)6]
(21
+
1)!
~
cd
X
exp[-(~b
+
D¢)(R]
+
1/2(-1)'~c
'(¢b
+
D-)"3
[(Z
+
b
+
c)l2
-(Z
-
tb-
-21
exp
[-(b
+
Pc)R
]-
/2
dZ
exp(-ZR)Z-2
(
d-Z)l
X
Pl
(1
d4
~)
te-aR-21
(E21[(¢b
+
rc
+
Z)6]
-
E21[(tb
+
c
-
Z)
G)
-/2[(21-
1)!]-'
exp
(ZR)
(-(rb
+
r,
+
Z)-'E2[(Ib
+
rc
+
Z)R]
+
El
[(~b
+
Dc+
Z)R]I
d-Z
Z-2
(z
d-z
Z-1
1c
(
d
)r-
X
(¢b
+
RC
+
Z)21-1+
'/2[(21
-
1)!]-l
exp(-ZR)
((rb
+
re-Z)-
T2
X
[(b
+
r
-
Z)R]
+
El
[(b
+
vc
-Z)R]
-
Z-2
(
-
Z
(
d-Y
i-i
M+1
(-m(N
+
1)
I(b
+
Z)21-1
-1/2
E
EZ
23
_
Z-3-2
+
)2N-2M-3E
1
M=O
N-O
23M!(l
_
M
1)!
(32N
X
(~b
+
~,c
)lN
-W
_ElM2N+4[(~b
+
~c)R]
~c
(-
-)
~c'(~b
+
~c2m
1
(7)
VOL.
58,
1967
35
CHEMISTRY:
H.
J.
SILVERSTONE
2__1_
(1
d
-
~
I2(l)
=
211!
(-1)I
Z-3
i-
-J
d
-
c
1
-[1
+
(rb-c
X
exp[-(f
R]
(rb
+
.c)
-
-
[1
+
(ab
+
tj
CR
]eXp[
(fb
+
ma)I]})
+
1/4(1)Z
d
exp(-ZR)Z-2
(
d
Z-1
l
1
d¶)l
)
C-l(-22
X
(P21[(
I,
-
c
+
Z)
R]
-
[t21[(fb
-
c
Z-
-
]
21A[(b
+
tc
+
Z)
R]
+
A21
[
(b
+
rc
-
Z)&
])
*
(8)
Horrendous
as
it
may
be,
the
analytical
formula
(eqs.
(6)-(8))
has
much
to
recom-
mend
it:
(1)
Computational
accuracy
with
analytical
formulas
is
easier
to
assess
than
with
numerical
integrations.
(2)
The
internuclear
angular
coordinates
enter
in
a
transparent
manner.
(3)
The
functions
g
and
XC1
are
characteristic
of
the
two-
center
charge
distribution
and
are
independent
of
the
integral
in
which
they
appear.
(4)
Convenient
numerical
methods
exist'
for
evaluating
the
classical
functions
g1,
JC1,
and
E21,
for
all
ranges
of
their
arguments
and
parameters.
(5)
Behavior
for
large
R
is
easy
to
estimate.
More
complicated
three-
and
four-center
integrals
work
out
similarly,
but
with
considerably
more
bookkeeping.
In
a
subsequent
paper,
detailed
derivations
and
computationally
convenient
formulas
will
be
given
for
the
general
rl2-'
integrals
involving
arbitrary
STO's.
*
Supported
by
a
National
Science
Foundation
grant.
'Shavitt,
I.,
in
Methods
in
Computational
Physics,
ed.
B.
Alder,
S.
Fernbach,
and
M.
Roten-
berg
(New
York:
Academic
Press
Inc.,
1963),
vol.
2,
p.
1.
2
Barnett,
M.
P.,
in
Methods
in
Computational
Physics,
ed.
B.
Alder,
S.
Fernbach,
and
M.
Rotenberg
(New
York:
Academic
Press
Inc.,
1963),
vol.
2,
p.
95.
3
Harris,
F.
E.,
and
H.
H.
Michels,
J.
Chem.
Phys.,
43,
S165
(1965).
4
Wahl,
A.
C.,
"The
evaluation
of
multicenter
integrals
by
polished
brute
force
techniques,"
paper
delivered
at
the
Slater
Symposium,
Sanibel
Island,
Florida,
January
16-21,
1967.
6
Barker,
R.
S.,
and
H.
Eyring,
J.
Chem.
Phys.,
22,
114,
1177
(1954).
6
Prosser,
F.
P.,
and
C.
H.
Blanchard,
J.
Chem.
Phys.,
36,
1112
(1962).
7Geller,
M.,
J.
Chem.
Phys.,
39,
853
(1963).
8
Silverstone,
H.
J.,
J.
Chem.
Phys.,
45,
4337
(1966).
9
Barnett,
M.
P.,
and
C.
A.
Coulson,
Phil.
Trans.
Roy.
Soc.
(London),
243,
221
(1951).
10
See,
e.g.,
Carrier,
G.
F.,
M.
Krook,
and
C.
E.
Pearson,
Functions
of
a
Complex
Variable
(New
York:
McGraw-Hill
Book
Co.,
Inc.,
1966).
11
Handbook
of
Mathematical
Functions,
ed.
M.
Abramowitz
and
I.
A.
Stegun
(National
Bureau
of
Standards
(U.S.),
Applied
Mathematics
Series,
no.
55,
1964).
12
The
author
has
been
unable
to
find
a
standard
notation
for
these
two
functions.
Throughout
this
paper,
all
"derivatives"
differentiate
everything
to
their
right.
36
PRoc.
N.
A.
S.
Chapter
Recent progress on computer evaluation of three-center integrals of 1/r12 with Slater-type orbitals is described. Fourier-transform-based formulas are first put in a programmable form. Then the special functions that occur are evaluated by either explicit expressions or recursion formulas. Convergence rate and asymptotics are touched upon briefly.
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The three‐center integral of r12− 1, with one electron on one center in a Slater‐type orbital, the second electron on two centers described by a product of Slater‐type orbitals, is evaluated analytically. The result is an infinite sum in which the internuclear angles appear in spherical harmonics and the internuclear distances appear in modified spherical Bessel functions and exponential‐type integrals. When one of the internuclear distances goes to zero, the two‐center hybrid integral is obtained as a finite sum. The main mathematical techniques used to evaluate the integrals are the Fourier‐transform convolution theorem, expansion of a Slater‐type orbital on one center about another, coupling properties of spherical harmonics, and contour integration.
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The relativistic Hartree-Fock-Roothaan (RHFR) formalism for closed-shell molecules is given. The wavefunction for such systems is taken as a single Slater determinant of 4-component molecular spinors (MS), where each MS is written as a linear combination of atomic spinors (LCAS/MS). The radial part of the atomic spinor (AS) is expanded in terms of Slater-type basis functions (STBF). The relativistic electronic Hamiltonian for the molecular system (in Born-Oppenheimer approximation) is the sum of Dirac Hamiltonians plus the interelectronic Coulomb repulsion and the magnetic part of the Breit interaction, but the retardation term is neglected at present. The reduction of the matrix elements of the relativistic Hamiltonian in terms of the nonrelativistic-type matrix elements is shown for any molecular system. Expressions for the matrix elements of the above-mentioned relativistic Hamiltonian are given for diatomics.
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Integrals over Slater atomic orbitals of the type ∫φa(r) pin φb(f) d3r, where pi is a component of the linear momentum operator, may be evaluated analytically by transforming the orbitals to momentum space. The procedure is an application of the Fourier convolution theorem. General formulae are given and uses for these integrals are suggested.
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Expansion of ψ(r)=ψ(r)YLM(θ,ϕ) in terms of spherical harmonics and radial functions, whose coordinates are measured from an arbitrary point in space, is obtained by use of the Fourier‐transform convolution theorem. For a specific ψ(r), two integrals most be evaluated to determine the expansion explicitly: (1) the radial part (k) of the Fourier transform of ψ(r); and (2) an integral of (k) with spherical Bessel functions. The examples of noninteger‐n and integer‐n Slater‐type orbitals are worked out by contour integration.
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We describe a computational scheme devised for using Fourier-transform-based analytic formulas for three-center integrals of r, wherein each electron is described by a two-center product of Slater-type orbitals. The asymptotic behavior of the auxiliary functions, which are related to modified spherical Bessel functions and to exponential integrals, is investigated, and recursive computational schemes are derived that are shown to be numerically stable for high summation indices and large internuclear distances. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
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Consideration has been given to two of the electronic repulsion integrals arising in the quantum‐mechanical energy calculation of the triatomic hydrogen molecular complex when no restrictions are placed on the effective nuclear charges of the composite atomic orbitals. The formulation of the integrals is first developed on a general basis and then restricted to the H3 molecule. The possibility of extension of the mathematical processes to similar electron interaction integrals in more complex molecules is pointed out. The integration formulas are given as rapidly converging series.
  • M P B Barnett
  • S Alder
  • M Fernbach
  • Rotenberg
Barnett, M. P., in Methods in Computational Physics, ed. B. Alder, S. Fernbach, and M. Rotenberg (New York: Academic Press Inc., 1963), vol. 2, p. 95. 3 Harris, F. E., and H. H. Michels, J. Chem. Phys., 43, S165 (1965).
The evaluation of multicenter integrals by polished brute force techniques," paper delivered at the Slater Symposium
  • A C Wahl
Wahl, A. C., "The evaluation of multicenter integrals by polished brute force techniques," paper delivered at the Slater Symposium, Sanibel Island, Florida, January 16-21, 1967. 6 Barker, R. S., and H. Eyring, J. Chem. Phys., 22, 114, 1177 (1954).
  • F P Prosser
  • C H Blanchard
Prosser, F. P., and C. H. Blanchard, J. Chem. Phys., 36, 1112 (1962). 7Geller, M., J. Chem. Phys., 39, 853 (1963).
  • H J Silverstone
Silverstone, H. J., J. Chem. Phys., 45, 4337 (1966).
  • M P Barnett
  • C A Coulson
Barnett, M. P., and C. A. Coulson, Phil. Trans. Roy. Soc. (London), 243, 221 (1951).
  • F E Harris
  • H H Michels
Harris, F. E., and H. H. Michels, J. Chem. Phys., 43, S165 (1965).
The evaluation of multicenter integrals by polished brute force techniques
  • A C Wahl
Wahl, A. C., "The evaluation of multicenter integrals by polished brute force techniques," paper delivered at the Slater Symposium, Sanibel Island, Florida, January 16-21, 1967.
  • F P Prosser
  • C H Blanchard
Prosser, F. P., and C. H. Blanchard, J. Chem. Phys., 36, 1112 (1962). 7Geller, M., J. Chem. Phys., 39, 853 (1963).