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:
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 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).
***,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.