MOLPRO

20 THE CI PROGRAM 164
problem is to replace the reference wave function Ψ (n)
ref
by the the relaxed reference functions
rlx
= ∑ RC Rn Φ
√ R
, (53)
∑ R CRn
2
Ψ (n)
which simply leads to
c 2 n = ∑CRn. (54)
R
Alternatively, one can linearly combine the fixed reference functions to maximize the overlap
with the MRCI wave functions. This yields projected functions
with
Ψ (n)
prj
= ∑
|Ψ (m)
ref 〉〈Ψ(m) ref |Ψ(n)
m
d mn = 〈Ψ (m)
ref |Ψ(n)
mrci 〉 = ∑
R
mrci 〉 = ∑
|Ψ (m)
m
These projected functions are not orthonormal. The overlap is
ref 〉d mn (55)
C (0)
Rm C Rn. (56)
〈Ψ (m)
prj |Ψ(n) prj 〉 = (d† d) mn . (57)
Symmetrical orthonormalization, which changes the functions as little as possible, yields
Ψ (n)
rot = ∑
|Ψ (m)
m
ref 〉u mn, (58)
u = d(d † d) −1/2 . (59)
The overlap of these functions with the MRCI wave functions is
〈Ψ (m)
rot |Ψ (n)
mrci 〉 = [(d† d)(d † d) −1/2 ] mn = [(d † d) 1/2 ] mn . (60)
Thus, in this case we use for the Davidson correction
c n = [(d † d) 1/2 ] nn . (61)
The final question is which reference energy to use to compute the correlation energy used in
eq. (50). In older MOLPRO version (to 2009.1) the reference wave function which has the
largest overlap with the MRCI wave function was used to compute the reference energy for the
corresponding state. But this can lead to steps of the Davidson corrected energies if the order
of the states swaps along potential energy functions. In the current version (2010.1 with patch
davidson) there are two options: the default is to use for state n the reference energy n, cf. eq.
(52) (assuming the states are ordered according to increasing energy). The second option is to
recompute the correlation energies using the rotated reference functions
E (n)
corr = E (n)
MRCI − 〈Ψ(n) rot |Ĥ|Ψ (n)
rot 〉 (62)