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quasi-diabatization [2022/08/16 14:15] – created - external edit 127.0.0.1quasi-diabatization [2024/01/08 13:24] (current) – external edit 127.0.0.1
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 +====== 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, [[https://dx.doi.org/10.1063/1.479214|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 [[the MCSCF program MULTI#diabatic orbitals|diabatic orbitals]]. 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.
 +
 +<code - 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
 +</code>
 +
 +This calculation produces the following results:
 +
 +<code>
 + 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
 +</code>
 +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:
 +
 +<code>
 + 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
 +</code>
 +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).
 +
 +<code - 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
 +</code>
 +
 +The calculation produces the following table
 +
 +<code>
 + 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
 +</code>
 +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.