# Orbital merging

Orbitals can be manipulated using the MERGE facility. For instance, this allows the construction of molecular orbitals from atomic orbitals, to merge and orthogonalize different orbital sets, or to perform $2 \times 2$ rotations between individual orbitals. Other orbital manipulations can be performed using the LOCALI program (see section orbital localization) or the MATROP program (section matrix operations).

The merge program is called using

MERGE [,namout.file]

All subcommands described in the following sections may be abbreviated by three characters. namout.file specifies the output data set (see also SAVE command). If namout.file is omitted and no SAVE card is present, the new orbitals are not saved. All output orbitals must be supplied via ORBITAL and ADD, MOVE, REPLACE, EXTRA, or  PROJECT directives before they can be saved.

ORBITAL,namin.file,specifications

Reads an input orbital set from a dump record. specifications can be used to select specific orbital sets, as described in section selecting orbitals and density matrices (ORBITAL, DENSITY). Subsets of these orbitals can be added to the output set by the ADD, MOVE, or EXTRA commands.

MOVE,orb1.sym1,orb2.sym2,orb3.sym3,ioff,fac,istart,iend

Moves orbitals orb1.sym1 to orb2.sym2 from the input set to the first vector of symmetry sym3 in the output set which is undefined so far. The first orb3-1 vectors in the output set are skipped regardless of whether they have been defined before or not. If sym2 $>$ sym1, sym3 will run from sym1 to sym2 and the input for sym3 has no effect. If orb1.sym1 is negative, abs(orb1) is the maximum number of orbitals to be moved, starting with orbital 1.sym1, up to orb2.sym2. If orb2.sym2 is negative, abs(orb2) is the maximum number of vectors to be moved, starting at orb1.isym1 up to the last orbital in symmetry sym2.

Orbitals from the input set which have already been moved or added to the output set are generally skipped. If orb1 and orb2 are zero, the whole input set is moved to the output set. In this case the input and output dimensions must be identical. If orb1 is nonzero but orb2 is zero, orb2 is set to the last orbital in symmetry sym2. If sym2=0, sym2 is set to sym1. ioff is an offset in the output vector, relative to the global offset set by OFFSET directive. fac has no effect for move. The elements istart to iend of the input vector are moved. If istart=0 and iend=0, the whole input vector is moved.

The usage of the MOVE directive is most easily understood by looking at the examples given below. See also ADD and EXTRA commands.

ADD,orb1.sym1,orb2.sym2,orb3.sym3,ioff,fac,istart,iend

This adds orbitals orb1.sym1 to orb2.sym2 to the output vectors, starting at orb3.sym3. The input vectors are scaled by the factor fac. If fac=0, fac is set to 1.0. For other details see MOVE command. Note, however, that the output vectors which have already been defined are not skipped as for MOVE.

REPLACE,orb1.sym1,orb2.sym2,orb3.sym3,ioff,,iustart,iend Replaces orbitals in the outputs set. If the output orbitals already exists, they are set to zero and then replaced by the given input orbitals. Otherwise as ADD.

See also MOVE and EXTRA commands.

EXTRA,exsym,orb1.sym1,orb2.sym2,orb3.sym3,ioff,fac,istart,iend

Works exactly as MOVE, but only input vectors with extra symmetry exsym are considered. If orb1.sym1 and orb2.sym2 are zero, all input vectors are moved to the output set ordered according to increasing extra symmetries.

Examples:

• EXTRA,1,-4.1 will move the next 4 orbitals in symmetry 1 which have extra symmetry 1. Orbitals which have been moved before are skipped.
• EXTRA,2,1.1 will move all orbitals of symmetry 1 which have extra symmetry 2. Orbitals which have been moved before are skipped.
• EXTRA will move all orbitals (all symmetries) and order them according to extra symmetries.
• EXTRA,3,1.1,0.8 Will move all orbitals which have extra symmetry 3 in all symmetries. Orbitals which have been moved before are skipped.

See also ADD and MOVE commands.

OFFSET,iof$_1$,iof$_2$,$\ldots$,iof$_8$;

Sets offsets in the output vector for symmetries 1 to 8. In subsequent MOVE or ADD commands, the input vectors are moved to the locations iof$_i$+1 in the output vectors. The offset for individual ADD or MOVE commands can be modified by the parameter ioff on these cards. This card should immediately follow the orbital directive to which it applies. Generally, this card is only needed if the dimensions of input and output vectors are not identical.

If the dimensions of the input orbital sets are smaller than the current basis dimension, the offsets are determined automatically in the following way: each time an orbital set is read in, the previous input orbital dimensions are added to the offsets. Hence, this works correctly if the orbital sets are given in the correct order and if the individual dimensions add up to the current total dimension. If this is not the case, the offsets should be specified on an OFFSET card which must follow the orbital directive.

PROJECT,namin.file

This command will read vectors from record namin.file. These vectors must have the same dimension as those of the current calculation. All orbitals defined so far by the ORBITAL, MOVE, and ADD directives are projected out of the input set. The projected orbitals are then orthonormalized and moved to the undefined output vectors. This should always yield a complete set of vectors.

ORTH,$n_1,n_2,\ldots,n_8$

Symmetrically orthonormalizes the first $n_i$ vectors in each symmetry $i$. These vectors must be supplied before by ORBITAL and MOVE or ADD directives.

SCHMIDT,$n_1,n_2,\ldots,n_8$

Schmidt orthonormalizes the first $n_i$ vectors in each symmetry $i$. These vectors must be supplied before by ORBITAL and MOVE or ADD directives.

ROTATE,iorb1.sym,iorb2,angle

Will perform $2 \times 2$ rotation of orbitals iorb1 and iorb2 in symmetry sym by the specified angle (in degree). angle=0 means to swap the orbitals (equivalent to angle=90) These vectors must be supplied before by ORBITAL and MOVE or ADD directives.

INIT,namout.file

Will initialize a new output set. All previous vectors in the output set are lost unless they have been saved by a SAVE directive!

SAVE,namout.file

Saves the current output set to record namout.file. The current output set must be complete and will be Schmidt orthonormalized before it is saved. If the SAVE directive is not supplied, the output vectors will be saved after all valid commands have been processed to the record specified on the MERGE card.

PRINT,iprint,ideb

Specifies print options.

• $iprint = 0$ no print
• $iprint \ge 1$: orthonormalized orbitals specified on ORTH card are printed.
• $iprint \ge 2$: orbitals are also printed before this orthonormalization.
• $iprint \ge 3$: all final vectors are printed.
• $ideb \ne 0$: the overlap matrices are printed at various stages.

This example merges the orbitals of H$_2$ and F

examples/h2f_merge.inp
***,example for merge
print,orbitals,basis
rh2=1.4
rhf=300.
basis=vdz
symmetry,x,y                            !use C2v symmetry
geometry={F}

text,F
{rhf;wf,9,1,1;occ,3,1,1;orbital,2130.2} !rhf for f-atom

text,H2
symmetry,x,y                            !use C2v symmetry
geometry={
H1,
H2,H1,rh2}

{hf;orbital,2100.2}                     !scf for h2
{multi;occ,2;orbital,2101.2}            !mcscf for h2

text,FH2
geometry={F;                            !linear geometry for F+H2
H1,F,rhf
H2,H1,rh2,F,180}

int
{merge
orbital,2130.2                          !rhf orbitals for F-atom
move,1.1,2.1,1.1                        !move orbitals 1.1, 2.1
move,3.1,0.4,4.1;                       !move all remaining, starting at 4.1
orbital,2100.2                          !hf orbitals for H2
move,1.1,0.4                            !move these to free positions
save,2131.2}                            !save merged orbitals

{rhf;occ,4,1,1;start,2131.2             !rhf for F+H2
orbital,2132.2}

{merge
orbital,2130.2                          !rhf orbitals for F-atom
move,1.1,2.1,1.1                        !move orbitals 1.1, 2.1
move,3.1,3.1,4.1;                       !move orbital 3.1 to 4.1
move,4.1,0.4,6.1                        !move all remaining, starting at 6.1
orbital,2101.2                          !mcscf orbitals for H2
move,1.1,0.4                            !move these to free positions
save,2141.2}                            !save merged orbitals

{multi;occ,5,1,1;start,2141.2}          !casscf for F+H2 using valence space

This example merges the SCF orbitals of N and O to get a full valence space for NO. In the simplest case the atomic calculations are performed in the individual separate basis sets, but using the same symmetry (C$_{2v}$) as the molecular calculation.

examples/no_merge1.inp
***,NO merge
r=2.1

symmetry,x,y
geometry={n}         !N-atom, c2v symmetry

{rhf;occ,3,1,1;      !rhf nitrogen
wf,7,4,3;            !4S state
orbital,2110.2}      !save orbitals to record 2110 on file 2

symmetry,x,y
geometry={o}

{rhf;occ,3,1,1;      !rhf for oxygen
wf,8,4,2             !3P state
orbital,2120.2}      !save orbitals to record 2120 on file 2

geometry={n;o,n,r}   ! NO molecule, c2v symmetry

int
{MERGE
ORBITAL,2110.2       ! read orbitals of N atom
MOVE,1.1,1.1         ! move 1s orbital to output vector 1.1
MOVE,2.1,2.1,3.1     ! move 2s orbital to output vector 3.1
MOVE,3.1,3.1,5.1     ! move 2pz orbital to output vector 5.1
MOVE,1.2,1.2         ! move 2px orbital to output vector 1.2
MOVE,1.3,1.3         ! move 2py orbital to output vector 1.3
MOVE,4.1,,7.1        ! move virtual orbitals of symmetry 1
MOVE,2.2,,3.2        ! move virtual orbitals of symmetry 2
MOVE,2.3,,3.3        ! move virtual orbitals of symmetry 2
MOVE,1.4             ! move virtual orbitals of symmetry 2
ORBITAL,2120.2       ! read orbitals of O atom
MOVE,1.1,0.4         ! move all oxygen orbitals into place
ROT,3.1,4.1,45;      ! rotate 2s orbitals to make bonding and antibonding
! linear combinations
ROT,5.1,6.1,-45;     ! rotate 2pz orbitals to make bonding and antibonding
! linear combinations
PRINT,1              ! set print option
ORTH,6,2,2           ! symmetrically orthonormalize the valence orbitals
! the resulting orbitals are printed
save,2150.2}         ! save merged orbitals to record 2150.2

{multi;occ,6,2,2     ! perform full valence casscf for NO
wf,15,2,1            ! 2Pix state
wf,15,3,1            ! 2Piy state
start,2150.2}        ! start with merged orbitals

One can also do the atomic calculations in the total basis set, using dummy cards. In this case the procedure is more complicated, since the union of the two orbital spaces is over-complete. The calculation can be done as follows:

a) SCF for the total molecule, orbitals saved to 2100.2

b) SCF for the N atom with dummy basis on the O atom, orbitals saved on 2110.2

c) SCF for the O atom with dummy basis on the N atom, orbitals saved on 2120.2

d) Merge the atomic SCF orbitals. Finally, obtain the virtual orbitals by projecting the merge orbitals out of the SCF orbitals for NO.

examples/no_merge2.inp
***,NO merge
geometry={n;o,n,r}
r=2.1

{rhf;occ,5,2,1     !rhf for NO
wf,15,2,1          !2Pi state
orbital,2100.2}    !save orbitals to record 2100 on file 2

dummy,o            !oxygen is dummy
{rhf;occ,3,1,1;     !rhf nitrogen
wf,7,4,3;          !4S state
orbital,2110.2}    !save orbitals to record 2110 on file 2

dummy,n            !nitrogen is dummy
{rhf;occ,3,1,1;    !rhf for oxygen
wf,8,4,2           !3P state
orbital,2120.2}    !save orbitals to record 2120 on file 2

dummy              ! remove dummies
{MERGE              !call merge program

ORBITAL,2110.2     ! read orbitals of N atom
MOVE,1.1,1.1       ! move input vector 1.1 to output vector 1.1
MOVE,2.1,3.1,3.1   ! move input vectors 2.1,3.1 to output vectors
! 3.1 and 4.1
MOVE,1.2,1.2       ! move input vector 1.2  to output vector 1.2
MOVE,1.3,1.3       ! move input vector 1.3  to output vector 1.3
ORBITAL,2120.2     ! read orbitals of O atom
MOVE,1.1,3.1       ! move input vectors 1.1 to 3.1 to output vectors
! 2.1, 5.1, 6.1
MOVE,1.2,1.2       ! move input vector 1.2 to output vector 2.2
MOVE,1.3,1.3       ! move input vector 1.3 to output vector 2.3
ROT,3.1,5.1,45;    ! rotate 2s orbitals to make bonding and antibonding
! linear combinations
ROT,4.1,6.1,-45;   ! rotate 2pz orbitals to make bonding and antibonding
! linear combinations
PRINT,1            ! set print option
ORTH,6,2,2         ! symmetrically orthonormalize the valence orbitals
! the resulting orbitals are printed
PROJ,2100.2        ! Project valence orbitals out of scf orbitals of the
! molecule and add virtual orbital set.
SAVE,2150.2        ! save merged orbitals to record 2150 on file 2
}

{multi;occ,6,2,2   ! perform full valence casscf for NO
wf,15,2,1          ! 2Pi state
wf,15,3,1          ! 2Pi state
start,2150.2}      ! start with merged orbitals