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 and states of HS, C 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 . Both states now belong to the 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 elements contain the state overlap matrix (bra index rans fastest).
`UMAT`- The first elements contain the transformation matrix.
`HDIA`- The first 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 reference geometry, and then at displaced geometries.

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).

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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molpro@molpro.net 2017-12-14