# 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 property 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$_{2v}$ geometries 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 diabatic orbitals. Secondly, the `DDR`

procedure can be used to find the transformation among the CI vectors.

The following input is required:

calls the DDR procedure.`DDR`

`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.The given states are included in the diabatization.`MIXING`

,*state1, state2, …*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`ENERGY`

,*e1, e2, …*`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:

The first $nstate \times nstate$ elements contain the state overlap matrix (bra index rans fastest).`SMAT`

The first $nstate \times nstate$ elements contain the transformation matrix.`UMAT`

The first $nstate \cdot (nstate+1)/2$ elements contain the lower triangle of the diabatic hamiltonian.`HDIA`

Non-adiabatic mixing angle in degree. This is available only in the two-state case.`MIXANG`

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.

- examples/h2s_diab1.inp
***,h2s Diabatization gprint,orbitals,civector symmetry,x orient,noorient !noorient should always be used for diabatization geometry={ s; h1,s,r1; h2,s,r2,h1,theta} basis=avdz !This basis is too small for real application r1=2.5 !Reference geometry theta=[92] r=[2.50,2.55,2.60] !Displaced geometries reforb=2140.2 !Orbital dumprecord at reference geometry refci=6000.2 !MRCI record at reference geometry savci=6100.2 !MRCI record at displaced geometries text,compute wavefunction at reference geometry (C2v) r2=r1 {hf;occ,9,2;wf,18,2,4; orbital,2100.2} {multi;occ,9,2;closed,4,1; wf,18,2;state,2; !1B1 and 1A2 states natorb,reforb !Save reference orbitals on reforb noextra} !Dont use extra symmetries {ci;occ,9,2;closed,4,1; !MRCI at reference geometry wf,18,2,0;state,2; !1B1 and 1A2 states orbital,reforb !Use orbitals from previous CASSCF save,refci} !Save MRCI wavefunction Text,Displaced geometries do i=1,#r !Loop over different r values data,truncate,savci+1 !truncate dumpfile after reference r2=r(i) !Set current r2 {multi;occ,9,2;closed,4,1; wf,18,2,0;state,2; !Wavefunction definition start,reforb !Starting orbitals orbital,3140.2; !Dump record for orbitals diab,reforb !Generate diabatic orbitals relative to reference geometry noextra} !Dont use extra symmetries {ci;occ,9,2;closed,4,1; wf,18,2,0;state,2; !1B1 and 1A2 states orbital,diabatic !Use diabatic orbitals save,savci} !Save MRCI for displaced geometries e1(i)=energy(1) !Save adiabatic energies e2(i)=energy(2) {ci;trans,savci,savci !Compute transition densities at R2 dm,7000.2} !Save transition densities on this record {ci;trans,savci,refci; !Compute transition densities between R2 and R1 dm,7100.2} !Save transition densities on this record {ddr density,7000.2,7100.2 !Densities for <R2||R2> and <R2||R1> orbital,3140.2,2140.2 !Orbitals for <R2||R2> and <R2||R1> energy,e1(i),e2(i) !Adiabatic energies mixing,1.2,2.2} !Compute mixing angle and diabatic energies mixci(i)=mixangci(1) !Mixing angle obtained from ci vectors only h11ci(i)=hdiaci(1) !Diabatic energies obtained from ci vectors only h21ci(i)=hdiaci(2) h22ci(i)=hdiaci(3) mixtot(i)=mixang(1) !Mixing angle from total overlap (including first-order correction) h11(i)=hdia(1) !Diabatic energies obtained from total overlap h21(i)=hdia(2) h22(i)=hdia(3) {table,r,e1,e2,h11ci,h22ci,h21ci,mixci title,Diabatic energies for H2S, obtained from CI-vectors format,'(f10.2,5f14.8,f12.2)' sort,1} {table,r,e1,e2,h11,h22,h21,mixtot title,Diabatic energies for H2S, obtained from CI-vectors and orbital correction format,'(f10.2,5f14.8,f12.2)' sort,1} enddo !end loop over i

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).

- examples/h2s_diab2.inp
***,h2s Diabatization and NACME calculation gprint,orbitals,civector symmetry,x orient,noorient !noorient should always be used for diabatization geometry={ s; h1,s,r1; h2,s,r2,h1,theta} basis=avdz !This basis is too small for real application r1=2.5 !Reference geometry theta=[92] r=[2.55,2.60] !Displaced geometries dr=[0,0.01,-0.01] !Samll displacements for finite difference NACME calculation reforb1=2140.2 !Orbital dumprecord at reference geometry refci=6000.2 !MRCI record at reference geometry savci=6100.2 !MRCI record at displaced geometries text,compute wavefunction at reference geometry (C2v) r2=r1 {hf;occ,9,2;wf,18,2,4;orbital,2100.2} {multi;occ,9,2;closed,4,1; wf,18,2;state,2; !1B1 and 1A2 states natorb,reforb1 !Save reference orbitals on reforb1 noextra} !Dont use extra symmetries {ci;occ,9,2;closed,4,1; !MRCI at reference geometry wf,18,2,0;state,2; !1B1 and 1A2 states orbital,reforb1 !Use orbitals from previous CASSCF save,refci} !Save MRCI wavefunction Text,Displaced geometries do i=1,#r !Loop over different r values data,truncate,savci+1 !truncate dumpfile after reference reforb=reforb1 do j=1,3 !Loop over small displacements for NACME r2=r(i)+dr(j) !Set current r2 {multi;occ,9,2;closed,4,1; wf,18,2,0;state,2; !Wavefunction definition start,reforb !Starting orbitals orbital,3140.2+j; !Dumprecord for orbitals diab,reforb !Generate diabatic orbitals relative to reference geometry noextra} !Dont use extra symmetries reforb=3141.2 !Use orbitals for j=1 as reference for j=2,3 {ci;occ,9,2;closed,4,1; wf,18,2,0;state,2; orbital,diabatic !Use diabatic orbitals save,savci+j} !Save MRCI for displaced geometries eadia=energy !Save adiabatic energies for use in ddr if(j.eq.1) then e1(i)=energy(1) !Save adiabatic energies for table printing e2(i)=energy(2) end if {ci;trans,savci+j,savci+j; !Compute transition densities at R2+DR(j) dm,7000.2+j} !Save transition densities on this record {ci;trans,savci+j,refci; !Compute transition densities between R2+DR(j) and R1 dm,7100.2+j} !Save transition densities on this record {ci;trans,savci+j,savci+1; !Compute transition densities between R and R2+DR(j) dm,7200.2+j} !Save transition densities on this record {ddr density,7000.2+j,7100.2+j !Densities for <R2+DR||R2+DR> and <R2+DR||R1> orbital,3140.2+j,2140.2 !Orbitals for <R2+DR||R2+DR> and <R2+DR||R1> energy,eadia(1),eadia(2) !Adiabatic energies mixing,1.2,2.2} !Compute mixing angle and diabatic energies if(j.eq.1) then !Store diabatic energies for R2 (DR(1)=0) mixci(i)=mixangci(1) !Mixing angle obtained from ci vectors only h11ci(i)=hdiaci(1) !Diabatic energies obtained from ci vectors only h21ci(i)=hdiaci(2) !HDIA contains the lower triangle of the diabatic hamiltonian h22ci(i)=hdiaci(3) mixtot(i)=mixang(1) !Mixing angle from total overlap (including first-order correction) h11(i)=hdia(1) !Diabatic energies obtained from total overlap h21(i)=hdia(2) h22(i)=hdia(3) end if mix(j)=mixang(1) !Store mixing angles for R2+DR(j) enddo !End loop over j dchi(i)=(mix(3)-mix(2))/(dr(2)-dr(3))*pi/180 !Finite difference derivative of mixing angle {ddr density,7201.2,7202.2,7203.2 !Compute NACME using 3-point formula orbital,3141.2,3142.2,3143.2 states,2.2,1.2} nacmeci(i)=nacme {table,r,mixci,mixtot,dchi,nacmeci Title,Mixing angles and non-adiabatic coupling matrix elements for H2S format,'(f10.2,4f14.4)' sort,1 } {table,r,e1,e2,h11ci,h22ci,h21ci Title,Diabatic energies for H2S, obtained from CI-vectors format,'(f10.2,5f16.8)' sort,1} {table,r,e1,e2,h11,h22,h21 title,Diabatic energies for H2S, obtained from CI-vectors and orbital correction format,'(f10.2,5f16.8)' sort,1} enddo

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.