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ANALYTICAL
EVALUATION
OF
THREE
AND
FOURCENTER
INTEGRALS
OF
r12'
WITH
SLATERTYPE
ORBITALS*
BY
HARRIS
J.
SILVERSTONE
DEPARTMENT
OF
CHEMISTRY,
JOHNS
HOPKINS
UNIVERSITY,
BALTIMORE
Communicated
by
Paul
H.
Emmett,
May
11,
1967
Three
and
fourcenter
integrals
of
rl2'
with
Slatertype
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,
14
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
fourcenter
integrals,
and
a
particular
threecenter
integral
is
evaluated
as
an
example.
The
method
is
beguilingly
simple.
The
first
key
ingredient
is
the
Fourier
trans
form
convolution
theorem,68
which
automatically
reduces
the
sixdimensional
integration
to
a
threedimensional
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
twocenter
charge
distribution
to
the
sum
of
Fourier
transforms
of
onecenter
distributions.
The
technique
is
successful
because
(1)
the
angular
inte
gration
in
the
convolution
integral
is
over
spherical
harmonicstherefore
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
twocenter
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
rc13121
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
exponentialtype
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)'xl
sinh
x;
3C(x)
=
(x)1(x'd/dx)1xl
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,68'
1
we
obtain
I
=
1/47r
12Na2NbNtVfd~kF(k)G(k)k2
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)Z2
(
dZ)l
X
Pl
(1
d4
~)
teaR21
(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
dZ
Z2
(z
dz
Z1
1c
(
d
)r
X
(¢b
+
RC
+
Z)211+
'/2[(21

1)!]l
exp(ZR)
((rb
+
reZ)
T2
X
[(b
+
r

Z)R]
+
El
[(b
+
vc
Z)R]

Z2
(

Z
(
dY
ii
M+1
(m(N
+
1)
I(b
+
Z)211
1/2
E
EZ
23
_
Z32
+
)2N2M3E
1
M=O
NO
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
Z3
i
J
d

c
1
[1
+
(rbc
X
exp[(f
R]
(rb
+
.c)


[1
+
(ab
+
tj
CR
]eXp[
(fb
+
ma)I]})
+
1/4(1)Z
d
exp(ZR)Z2
(
d
Z1
l
1
d¶)l
)
Cl(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
fourcenter
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
1621,
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:
McGrawHill
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.