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42 QUASI-DIABATIZATION

The DDR procedure can also be used to generate quasi-diabatic states and energies for MRCI wavefucntions (CASSCF case can be treated as special case using the NOEXC directive in the MRCI). The quasi-diabatic states have the propery that they change as little as possible relative to a reference geometry; with other words, the overlap between the states at the current geometry with those at a reference geometry is maximized by performing a unitary transformation among the given states. Preferably, the adiabatic and diabatic states should be identical at the reference geometry, e.g., due to symmetry. For instance, in the examples given below for the $^1B_1$ and $^1A_2$ states of H$_2$S, C$_{2v}$ geomtries are used as reference, and at these geometries the states are unmixed due to their different symmetry. At the displaced geometries the molecular symmetry is reduced to $C_S$. Both states now belong to the $^1A''$ irreducible representation and are strongly mixed. For a description and application of the procedure described below, see D. Simah, B. Hartke, and H.-J. Werner, J. Chem. Phys. 111, 4523 (1999).

This diabatization can be done automatically and requires two steps: first, the active orbitals of a CASSCF calculation are rotated to maximize the overlap with the orbitals at the reference geometry. This is achieved using the DIAB procedure described in section 19.5.9. Secondly, the DDR procedure can be used to find the transformation among the CI vectors.

The following input is required:

DDR
calls the DDR procedure.
ORBITAL,orb1, orb2
orb1 and orb2 are the (diabatic) orbitals at the current and reference geometry, respectively.
DENSITY,trdm1,trdm2
trdm1 are the transition densities computed at the current geometry, trdm2 are transition densities computed using the wavefunctions of the current (bra) and reference (ket) geometries.
MIXING,state1, state2, ...
The given states are included in the diabatization.
ENERGY,e1, e2, ...
Adiabatic energies of the states. If this input card is present, the Hamiltonian in the basis of the diabatic states is computed and printed. Alternatively, the energies can be passed to DDR using the Molpro variable EADIA.

The results are printed and stored in the following Molpro variables, provided the ENERGY directive or the EADIA variable is found:

Results including the first-order orbital correction:

SMAT
The first $nstate \times nstate$ elements contain the state overlap matrix (bra index rans fastest).
UMAT
The first $nstate \times nstate$ elements contain the transformation matrix.
HDIA
The first $nstate \cdot (nstate+1)/2 $ elements contain the lower triangle of the diabatic hamiltonian.
MIXANG
Non-adiabatic mixing angle in degree. This is available only in the two-state case.

The corresponding results obtained from the CI-vectors only (without orbital correction) are stored in the variables [SMATCI], UMATCI, HDIACI, and MIXANGCI.

The way it works is most easily demonstrated for some examples. In the following input, the wavefunction is first computed at the $C_{2v}$ reference geometry, and then at displaced geometries.

h2s_diab1.com

This calculation produces the following results:

 Diabatic energies for H2S, obtained from CI-vectors

        R       E1            E2            H11CI         H22CI         H21CI          MIXCI
      2.50 -398.64296319 -398.63384782 -398.64296319 -398.63384782    0.00000000        0.00
      2.55 -398.64572746 -398.63666636 -398.64509901 -398.63729481   -0.00230207       15.27
      2.60 -398.64911752 -398.63771802 -398.64662578 -398.64020976   -0.00471125       27.87

 Diabatic energies for H2S, obtained from CI-vectors and orbital correction

        R       E1            E2            H11           H22           H21           MIXTOT
      2.50 -398.64296319 -398.63384782 -398.64296319 -398.63384782    0.00000000        0.00
      2.55 -398.64572746 -398.63666636 -398.64509941 -398.63729441   -0.00230139       15.26
      2.60 -398.64911752 -398.63771802 -398.64662526 -398.64021027   -0.00471160       27.88

The results in the first table are obtained from the CI-contribution to the state-overlap matrix only, while the ones in the second table include a first-order correction for the orbitals. In this case, both results are almost identical, since the DIAB procedure has been used to minimize the change of the active orbitals. This is the recommended procedure. If simply natural orbitals are used without orbital diabatization, the following results are obtained from the otherwise unchanged calculation:

 Diabatic energies for H2S, obtained from CI-vectors

        R       E1            E2            H11CI         H22CI         H21CI          MIXCI
      2.50 -398.64296319 -398.63384782 -398.64296319 -398.63384782    0.00000000        0.00
      2.55 -398.64572742 -398.63666630 -398.64475612 -398.63763760   -0.00280315       19.11
      2.60 -398.64911746 -398.63771803 -398.64521031 -398.64162518   -0.00541050       35.83

 Diabatic energies for H2S, obtained from CI-vectors and orbital correction

        R       E1            E2            H11           H22           H21           MIXTOT
      2.50 -398.64296319 -398.63384782 -398.64296319 -398.63384782    0.00000000        0.00
      2.55 -398.64572742 -398.63666630 -398.64509146 -398.63730226   -0.00231474       15.36
      2.60 -398.64911746 -398.63771803 -398.64648358 -398.64035190   -0.00480493       28.73

It is seen that the mixing obtained from the CI vectors only is now very different and meaningless, since the orbitals change significantly as function of geometry. However, the second calculations, which accounts for this change approximately, still gives results in quite good agreement with the calculation involving diabatic orbitals.

The final examples shows a more complicated input, which also computes the non-adiabatic coupling matrix elements. In a two-state model, the NACME should equal the first derivative of the mixing angle. In the example, the NACME is computed using the 3-point DDR method (NACMECI), and also by finite difference of the mixing angle (DCHI).

h2s_diab2.com

The calculation produces the following table

 Mixing angles and non-adiabatic coupling matrix elements for H2S

        R          MIXCI        MIXTOT          DCHI       NACMECI
      2.55       15.2694       15.2644       -5.2226       -5.2365
      2.60       27.8740       27.8772       -3.4702       -3.4794

 Diabatic energies for H2S, obtained from CI-vectors

        R       E1            E2            H11CI         H22CI         H21CI
      2.55 -398.64572746 -398.63666636 -398.64509901 -398.63729481   -0.00230207
      2.60 -398.64911752 -398.63771802 -398.64662578 -398.64020976   -0.00471125

 Diabatic energies for H2S, obtained from CI-vectors and orbital correction

        R       E1            E2            H11           H22           H21
      2.55 -398.64572746 -398.63666636 -398.64509941 -398.63729441   -0.00230139
      2.60 -398.64911752 -398.63771802 -398.64662526 -398.64021027   -0.00471160

As expected the coupling matrix elements obtained from the 3-point DDR calculation (NACMECI) and by differentiating the mixing angle (DCHI) are in close agreement.


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