20.7 Cluster corrections for multistate MRCI
In the following, we assume that
are the normalized reference and MRCI wave functions for state , respectively.
are the coefficients of the reference configurations in the initial reference functions
and are the relaxed coefficients of these configurations in the final MRCI wave function.
is the remainder of the MRCI wave function, which is orthogonal to all reference
configurations .
The corresponding energies are defined as
The standard Davidson corrected correlation energies are defined as
where is the coefficient of the (fixed) reference function in the MRCI wave function:



(42) 
and the correlation energies are



(43) 
In the vicinity of avoided crossings this correction may give unreasonable results
since the reference function may get a small overlap
with the MRCI wave function. One way to avoid this problem is to replace the reference wave function
by the the relaxed reference functions
which simply leads to
Alternatively, one can linearly combine the fixed reference functions to maximize the overlap
with the MRCI wave functions. This yields projected functions
with
These projected functions are not orthonormal. The overlap is
Symmetrical orthonormalization, which changes the functions as little as possible, yields
The overlap of these functions with the MRCI wave functions is
Thus, in this case we use for the Davidson correction
The final question is which reference energy to use to compute the correlation energy used in eq. (43). 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 this version
there are two options: the default is to use for state the reference energy , cf. eq. (45)
(assuming the states are ordered according
to increasing energy). The second option is to recompute the correlation energies using the rotated reference functions
Both should give smooth potentials (unless at conical intersections or crossings of states with different symmetries),
but there is no guarantee that the Davidson corrected energies of different states don't cross. This problem is
unavoidable for nonvariational energies. The relaxed and rotated Davidson corrections give rather similar results;
the rotated one yields somewhat larger cluster corrections and was found to give better results in the case of the
F + H potential [see J. Chem. Phys. 128, 034305 (2008)].
By default, the different cluster corrections listed in Table 9 are computed in multistate MRCI calculations.
and stored in variables.
Table 9:
Cluster corrections computed in multistate MRCI calculations. By default, the energies are
in increasing order of the MRCI total energy. In singlestate calculations only the fixed and relaxed values are available.
Name 
(Eq.) 
(Eq.) 
Variable 

Using standard reference energies: 

Fixed 
(44) 
(45) 
ENERGD1(n) 

Relaxed 
(47) 
(45) 
ENERGD0(n) 

Rotated 
(54) 
(45) 
ENERGD2(n) 

Using rotated reference energies: 

Relaxed 
(47) 
(55) 
ENERGD3(n) 

Rotated 
(54) 
(55) 
ENERGD4(n) 

By default, ENERGD(n)=ENERGD0(n). This can be changed by setting OPTION,CLUSTER=x;
then ENERGD(n)=ENERGD(n) (default ).
The behaviour of Molpro 2009.1 and older can be retrieved using
MRCI,SWAP,ROTREF=1.
molpro@molpro.net 20181210