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20.7 Cluster corrections for multi-state MRCI

In the following, we assume that

$\displaystyle \Psi_{\rm ref}^{(n)}$ $\textstyle =$ $\displaystyle \sum_R C_{Rn}^{(0)} \Phi_R$ (37)
$\displaystyle \Psi_{\rm mrci}^{(n)}$ $\textstyle =$ $\displaystyle \sum_R C_{Rn} \Phi_R + \Psi_c$ (38)

are the normalized reference and MRCI wave functions for state $n$, respectively. $C_R^{(0)}$ are the coefficients of the reference configurations in the initial reference functions and $C_{Rn}$ are the relaxed coefficients of these configurations in the final MRCI wave function. $\Psi_c$ is the remainder of the MRCI wave function, which is orthogonal to all reference configurations $\Phi_R$.

The corresponding energies are defined as

$\displaystyle E_{\rm ref}^{(n)}$ $\textstyle =$ $\displaystyle \langle \Psi_{\rm ref}^{(n)} \vert \hat H \vert \Psi_{\rm ref}^{(n)}\rangle,$ (39)
$\displaystyle E_{\rm mrci}^{(n)}$ $\textstyle =$ $\displaystyle \langle \Psi_{\rm mrci}^{(n)} \vert \hat H \vert \Psi_{\rm mrci}^{(n)}\rangle.$ (40)

The standard Davidson corrected correlation energies are defined as
$\displaystyle E^{n}_{\rm D}$ $\textstyle =$ $\displaystyle E_{\rm corr}^{(n)}\cdot \frac {1-c_n^2}{ c_n^2 }$ (41)

where $c_n$ is the coefficient of the (fixed) reference function in the MRCI wave function:
$\displaystyle c_n = \langle \Psi_{\rm ref}^{(n)}\vert\Psi_{\rm mrci}^{(n)} \rangle = \sum_R C_{Rn}^{(0)} C_{Rn},$     (42)

and the correlation energies are
$\displaystyle E_{\rm corr}^{(n)}=E_{\rm mrci}^{(n)}-E_{\rm ref}^{(n)} .$     (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 $\Psi_{\rm ref}^{(n)}$ by the the relaxed reference functions
$\displaystyle \Psi_{\rm rlx}^{(n)}$ $\textstyle =$ $\displaystyle \frac{\sum_R C_{Rn} \Phi_R}{\sqrt{\sum_R C_{Rn}^2}},$ (44)

which simply leads to
$\displaystyle c_n^2$ $\textstyle =$ $\displaystyle \sum_R C_{Rn}^2.$ (45)

Alternatively, one can linearly combine the fixed reference functions to maximize the overlap with the MRCI wave functions. This yields projected functions
$\displaystyle \Psi_{\rm prj}^{(n)}$ $\textstyle =$ $\displaystyle \sum_m \vert\Psi_{\rm ref}^{(m)}\rangle\langle \Psi_{\rm ref}^{(m...
...t\Psi_{\rm mrci}^{(n)} \rangle
= \sum_m \vert\Psi_{\rm ref}^{(m)}\rangle d_{mn}$ (46)

$\displaystyle d_{mn}$ $\textstyle =$ $\displaystyle \langle \Psi_{\rm ref}^{(m)}\vert\Psi_{\rm mrci}^{(n)} \rangle = \sum_R C_{Rm}^{(0)} C_{Rn}.$ (47)

These projected functions are not orthonormal. The overlap is
$\displaystyle \langle \Psi_{\rm prj}^{(m)} \vert \Psi_{\rm prj}^{(n)} \rangle$ $\textstyle =$ $\displaystyle ({\bf d}^{\dagger} {\bf d})_{mn}.$ (48)

Symmetrical orthonormalization, which changes the functions as little as possible, yields
$\displaystyle \Psi_{\rm rot}^{(n)}$ $\textstyle =$ $\displaystyle \sum_m \vert\Psi_{\rm ref}^{(m)}\rangle u_{mn},$ (49)
$\displaystyle {\bf u}$ $\textstyle =$ $\displaystyle {\bf d} ({\bf d}^{\dagger} {\bf d})^{-1/2}.$ (50)

The overlap of these functions with the MRCI wave functions is
$\displaystyle \langle \Psi_{\rm rot}^{(m)} \vert \Psi_{\rm mrci}^{(n)} \rangle$ $\textstyle =$ $\displaystyle [({\bf d}^{\dagger} {\bf d}) ({\bf d}^{\dagger} {\bf d})^{-1/2}]_{mn}
= [({\bf d}^{\dagger} {\bf d})^{1/2}]_{mn}.$ (51)

Thus, in this case we use for the Davidson correction
$\displaystyle c_n$ $\textstyle =$ $\displaystyle [({\bf d}^{\dagger} {\bf d})^{1/2}]_{nn}.$ (52)

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 $n$ the reference energy $n$, 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
$\displaystyle E_{corr}^{(n)}$ $\textstyle =$ $\displaystyle E_{\rm MRCI}^{(n)} - \langle \Psi_{\rm rot}^{(n)} \vert \hat H \vert \Psi_{\rm rot}^{(n)} \rangle$ (53)

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 non-variational 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$_2$ potential [see J. Chem. Phys. 128, 034305 (2008)].

By default, the different cluster corrections listed in Table 9 are computed in multi-state MRCI calculations. and stored in variables.

Table 9: Cluster corrections computed in multi-state MRCI calculations. By default, the energies are in increasing order of the MRCI total energy. In single-state calculations only the fixed and relaxed values are available.
Name $c_n$ (Eq.) $E_{corr}^{(n)}$ (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$x$(n) (default $x=0$). The behaviour of Molpro 2009.1 and older can be retrieved using


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