Page 43 - Valence Bond Methods. Theory and Applications

2H 2 and localized orbitalØ
26
s
where we hŁve writteð out all of the terms. The 1 function ið Eq. (2.10) is
normalized, sc

1s a (1)|1s a (1) =
1s b (2)|1s b (2) ,
(2.16)
=
1s b (1)|1s b (1) , (2.15)
=
1s a (2)|1s a (2) , (2.17)
= 1. (2.18)
The other four integrals are alsc equal tc one another, and this is a function of the
distance, R, betweeð the two atoms calleł the overlap integral, S(R). The overlap
integral is an elementary integral ið the appropriate coordinate system, confocal
ellipsoidal–hyperboloidal coordinates[27]‚ Ið terms of the function of Eq. (2.12) it
has the form
2
S(w) = (1 + w + w /3) exp(−w), (2.19)
w = αR, (2.20)

1
and we see that the normalization constant for ψ(1, 2) is
2
N = [2(1 + S )] −1/2 . (2.21)
1
We may now substitute ψ(1, 2) intc the Rayleigh quotient and obtaið an estimate
of the total energy,
1
1
E(R) =
ψ(1, 2)|H| ψ(1, 2) ≥ E 0 (R), (2.22)
where E 0 is the true ground state electronic energy for H 2 . This expression iðvolves
four new integrals that alsc can be evaluateł ið confocal ellipsoidal–hyperboloidal
coordinates. Ið this case all are not sc elementary, iðvolving, as they do, expan-
sions ið Legendre functions. The final energy expression is (α = 1 ið all of the
integrals)

j 1 (R) + S(R)k 1 (R) j 2 + k 2 1
E = 2h − 2 + + , (2.23)
1 + S(R) 2 1 + S(R) 2 R
where
2
h = α /2 − α, (2.24)
j 1 =−[1 − (1 + w)e −2w ]/R, (2.25)
k 1 =−α(1 + w)e −w , (2.26)
3
2
j 2 = [1 − (1 + 11wζ8 + 3w /4 + w /6)e −2w ]/R, (2.27)
2
2
k 2 ={6[S(w) (C + ln w) − S(−w) E 1 (4w) + 2S(w)S(−w)E 1 (2w)]
4
3
2
+ (25wζ8 − 23w /4 − 3w − w /3)e −2w }/(5R), (2.28)
2
l 2 = [(5/16 + wζ8 + w )e −w − (5/16 + wζ8)e −3w ]/R. (2.-)