# The SCF program

The Hartree-Fock self-consistent field program is invoked by one of the following commands:

• HF or RHF calls the spin-restricted Hartree-Fock program
• UHF or UHF-SCF,options calls the spin-unrestricted Hartree-Fock program

In contrast to older versions of molpro, the HF and RHF directives have identical functionality and can both be used for closed-shell or open-shell calculations. Other aliases are HF-SCF or RHF-SCF.

Often, no further input is necessary. By default, the number of electrons is equal to the nuclear charge, the spin multiplicity is 1 (singlet) for an even number of electrons and 2 (doublet) otherwise, and the wavefunction is assumed to be totally symmetric (symmetry 1) for singlet calculations. The Aufbau principle is used to determine the occupation numbers in each symmetry. Normally, this works well in closed-shell and many open-shell cases, but sometimes wrong occupations are obtained. In such cases, the OCC and/or CLOSED directives can be used to force convergence to the desired state. The default behaviour can be modified either by options on the command line, or by directives.

In open-shell cases, we recommend to use the WF, OCC, CLOSED, or OPEN cards to define the wavefunction uniquely. Other commands frequently used are START and ORBITAL (or SAVE) to modify the default records for starting and optimized orbitals, respectively. The  SHIFT option or directive allows to modify the level shift in the RHF program, and EXPEC to calculate expectation values of one-electron operators (see section One-electron operators and expectation values (GEXPEC)). Section handling difficult cases: When SCF does not converge discusses strategies for dealing difficult molecules and convergence problems. With molpro 2020 or later, it is recommended to use the new first-order mcscf program MULTI,SO-SCI in such cases. With Molpro 2021 or later, it is also possible to use HF,SO-SCI, which is equivalent to MULTI,SO-SCI, except that the Hartree-Fock default occupations and options are used. This option is not available for UHF.

Density fitting can be used for closed and open-shell (RHF and UHF) HF and is invoked by a prefix DF- (DF-HF, DF-RHF, or DF-UHF), see section density fitting). Density fitting very much speeds up calculations for large molecules and is strongly recommended. The greatest savings are seen for large basis sets with high angular momentum functions. The DF-HF program is well parallelized. Density fitting also works well for DF-HF,SO-SCI and Kohn-Sham (DF-KS or DF-UKS) calculations.

Using

LDF-HF, options
LDF-RHF, options
LDF-UHF, options

enables the new local density fitting Hartree-Fock program described in C. Köppl and H.-J. Werner, J. Chem. Theory Comput. 12, 3122 (2016) (LDF-HF and LDF-RHF are the same). For large and dense 3-dimensional systems this can speed up DF-HF calculations typically by a factor of 4-5, and even more if run in parallel or for extended 1- or 2-dimensional systems. Since the total HF energy is rather sensitive to the local fitting approximation, the final energy is either recomputed without local fitting (default), or with larger domains. Typically, with default options, $\mu$H accuracy of the absolute HF energy is achieved, and the effect of the approximations on relative energies is negligible. This is also true for subsequent local coupled-cluster calculations (see section local correlation methods with pair natural orbitals (PNOs)). The program is well parallelized and can also be run across several computer nodes.

The following options can be used to affect the local approximations:

• IDFDOM_LSCF=value Connectivity criterion for fitting domains (default 3).
• RDFDOM_LSCF=value Distance criterion for fitting domains (default 7 bohr).
• FITMOS_LSCF=thresh Threshold for LMO domains (default $10^{-6})$.
• FITENERGY=value If nonzero, recompute the final energy without local fitting (default). If set to zero, the domain parameters below are used for computing the final energy.
• DYNFIT=value If non-zero and FITENERGY=0, increase IDFDOM_LSCF by DYNFIT and RDFDOM_LSCF by 2*DYNFIT for computing the final energy (default 0).
• RDFDOM_LFINAL=value Connectivity criterion for fitting domains in the last iteration (default 6, only used if DYNFIT=0 and FITENERGY=0).
• IDFDOM_LFINAL=value Distance criterion for fitting domains in the last iteration

(default 13 bohr, only used if DYNFIT=0 and FITENERGY=0).

• FITMOS_LFINAL=thresh Threshold for LMO domains in the last iteration (default $10^{-8}$, only used if DYNFIT=0 and FITENERGY=0).
• ROTDIAG=iter Use fast rotational update of the orbitals, starting in iteration iter (default 3). Setting ROTDIAG=0 disables the method and uses full diagonalization of the Fock matrix.

Other options are as in the standard HF and DF-HF programs. Please refer section density fitting for more options regarding density fitting.

For more information about local fitting see R. Polly, H.-J. Werner, F. R. Manby, and Peter J. Knowles, Fast Hartree-Fock theory using local density fitting approximations, Mol. Phys. 102, 2311 (2004) and C. Köppl and H.-J. Werner, J. Chem. Theory Comput. 12, 3122-3134 (2016).

All publications resulting from LDF-HF or LDF-KS calculations should cite this work.

This program (available since Molpro2020.3) is called using

cahf,options

or, with density fitting,

df-cahf,options

It yields orbitals which are equivalent to the orbitals of a state-averaged CASSCF calculation averaging over all states of all possible spin-manifolds given by a chosen active space, where the states of different spin-manifolds are given weighting factors corresponding to their MS-degeneracy. It is also possible to define two or more shells of active orbitals, and the calculation is then equivalent to a state-averaged RASSCF (restricted active space) calculation in which the configuration space is the direct product of the given CASSCF subspaces.

The active orbitals always follow the closed-shell ones, and it is essential to start with a qualitatively correct set of active orbitals in order to converge to the desired solution. The AVAS procedure is often very helpful to define the active space, see below.

The active space must be defined in one of the following ways (or consistent combinations of these):

• Using OCC and CLOSED directives following the command line.
• By specifying the active orbitals explicitly using SHELL directives.

SHELL directives are mandatory if more than one active shell is used. For each shell the active orbitals are specified as

SHELL,nel,orbital list

where nel is the number of electrons in the active shell and orbital list is given in the form number1.sym1,number2.sym2,.... Orbital numbers in the same symmetry must be consecutive. A range of orbitals in the same symmetry can be given as number1.sym,-number2.sym, which means from number1 to number2. The lowest orbital in a symmetry isym on any SHELL directive must equal iclos(isym)+1, where iclos(isym) is the number of closed-shell orbitals in the symmetry as given on a CLOSED directive. The highest orbital in symmetry isym on any SHELL directive equals iocc(isym), where iocc(isym) is the total number of occupied orbitals, as defined on an OCC directive, and any orbitals in-between must be defined on the SHELL directives. IF OCC and/or CLOSED directives are not given, the number of occupied and closed-shell orbitals is derived from the SHELL directive(s). But this only works if all symmetries occur on them.

OPTIONS can be the same as described for HF or DF-HF. The most useful ones in the context of CAHF are

• SHIFTC=value: shift for closed shell orbitals (negative means shifting down), default -0.6 Hartree.
• SHIFTO=value: shift for open-shell (active) orbitals, default -0.3 Hartree.
• MINGAP=value: minimal energy gaps between closed-active and active-virtual orbitals (default 0.5 Hartree). If more than one shell is defined, an energy gap is ensured also between the shells. This parameter is essential for convergence of CAHF.

If a subsequent CASCI calculation using MULTI is performed using the CAHF orbitals, the corresponding state-averaged CASSCF wave functions for the individual states are obtained. The orbitals can also be employed in subsequent multi-reference correlation methods, such as CASPT2 and MRCI. The program is for example useful for the efficient calculation of transition-metal/lanthanide/actinide compounds, where many nearly degenerate spin-eigenstates are of interest.

Example:

 geometry={O}
basis=vdz
{cahf
occ,2,1,1,,1
closed,2}

Alternative:

examples/o_cahf.inp
 geometry={O}
basis=vdz
{cahf
closed,2
shell,4,1.2,1.3,1.5}

This yields the same orbitals as

{multi
occ,2,1,1,,1
closed,2
wf,spin=2,sym=4;weight,3
wf,spin=2,sym=6;weight,3
wf,spin=2,sym=7;weight,3 !3P states
wf,spin=0,sym=1;state,3  !1D and 1s states
wf,spin=0,sym=4          !1D
wf,spin=0,sym=6          !1D
wf,spin=0,sym=7          !1D
}

In order to converge to the correct solution, appropriate starting orbitals are essential. This is not always given by the default density guess. The AVAS procedure may be used to obtain proper starting orbitals. An example with two Ytterbium atoms is given below. Previous cahf calculations (with a smaller basis set and local density fitting approximations) can be found in Phys. Chem. Chem. Phys. 21, 9769 (2019), and the structure is from J. Am. Chem. Soc. 140, 2504–2513 (2018) (yb_hq_no3.xyz geometry file) The f-orbitals on each Yb atom form the two active shells. The AVAS orbitals are localised and ordered according to center numbers:

examples/Yb2.inp
memory,200m
geometry=yb_hq_no3.xyz

charge=-1
basis=def2-tzvpp

{avas,locorb=1,nela=26  ! 26 active electrons, localized orbitals
center,1,4f             ! This assumes that the first 2 atoms are Yb
center,2,4f}

{df-cahf
cahf,13,227.1,-233.1
cahf,13,234.1,-240.1}


Note that it is also possible to carry out such calculations using the MULTI program. Due to faster and more robust convergence this may be less expensive.

Using

LDF-CAHF

calls the local density fitting configuration-averaged Hartree-Fock (LDF-CAHF) described in P. P. Hallmen et. al, J. Chem. Phys. 147, 164101 (2017), and for multi-shell problems to Phys. Chem. Chem. Phys. 21, 9769 (2019). The further input is the same as for the non-local variant.

Considering accuracy, the same holds as for LDF-RHF. Local approximations cause an overhead in the density fitting calculation of the Fock matrices, and speedups can only be expected for rather large systems. For a molecule as shown in the last example for df-cahf, the computation times of local and non-local calculations are very similar, and it is not worthwhile to use the local version.

In this section the options for HF|RHF|UHF|CAHF are described. For further options affecting Kohn-Sham calculations see section the density functional program. For compatibility with previous MOLPRO versions, options can also be given on subsequent directives, as described in later sections.

• ACCU[RACY]=accu Convergence threshold for the density matrix (square sum of the density matrix element changes). If accu$> 1$, a threshold of $10^{-accu}$) is used. The default depends on the global ENERGY threshold, and the threshold is automatically tightened in geometry optimizations or frequency calculations (unless a tight threshold is given).
• ENERGY=thrden The convergence threshold for the energy. The default depends on the global ENERGY threshold.
• START=record Record holding start orbitals.
• SAVE|ORBITAL=record Dump record for orbitals.
• MAXIT=maxit Maximum number of iterations (default 60)
• SHIFTA|SHIFTC=shifta Level shift for closed-shell orbitals in RHF (default $-0.3$) and $\alpha$-spin orbitals in UHF (default 0).
• SHIFTB|SHIFTO=shiftb Level shift for open-shell orbitals in RHF and $\beta$-spin orbitals in UHF (default 0)
• NITORD|NITORDER=nitord Starting with iteration nitocc, the orbital occupation pattern is kept fixed: The orbitals are reordered after each iteration to obtain maximum overlap with the closed-shell/open-shell/virtual spaces from the previous iteration. This takes only effect after nitord iterations. The default depends on the quality of the starting guess.
• NITSH|NITSHIFT=nitsh If the iteration count is smaller than nitsh, the shifts are set to zero. The default depends on the quality of the starting guess.
• NITCL|NITCLOSED=nitcl If the iteration count is smaller than nitcl, only the closed-shell part of the Fock matrix is used (default nitcl$=0$). This option is left for compatibility purposes, it almost never helps.
• NITOCC=nitocc Starting with iteration nitocc the occupation pattern is kept fixed. The default depends on the quality of the starting guess.
• NITORT|NITORTH=nitort The orbitals are reorthonormalized after every nitort iterations. The default is nitort$=10$.

Note that in case of a restart the iteration count starts with 3.

The following options can be used to localize the RHF orbitals (not implemented for UHF):

• LOCORB=value if set to 1, the valence orbitals are localized at the end (default 0).
• LOC_METHOD=IBO|PM|BOYS|NPA determines localization method if LOCORB=1. Default is IBO.

Only the valence orbitals are localized, and the options only work without symmetry. Alternatively the LOCALI program can be used for localization, which offers more flexible options.

In calculations with very large basis sets, the diagonalization time becomes a significant fraction of the total CPU time. This can be reduced using the orbital rotation method as described in R. Polly, H.-J. Werner, F. R. Manby, and Peter J. Knowles, Mol. Phys. 102, 2311 (2004))

• MINROT=minrot If minrot$\ge 0$, the orbital rotation method is employed. Explicit diagonalization of the full Fock matrix is performed in the first minrot iterations and in the last iteration. If minrot=0, a default is used which depends on the starting guess.
• NEXPR=nexpr Number of terms used in the exponential expansion of the unitary orbital transformation matrix (default 4).
• DEROT=nexpr Energy gap used in the orbital rotation method. For orbitals within $\pm$derot hartree of the HOMO orbital energy the Fock matrix is constructed and diagonalized (default 1.0)
• JACOBI=jacobi If nonzero, use Jacobi diagonalization.

For more details, see IPOL directive.

• IPTYP=iptyp Interpolation type (default DIIS, see IPOL directive).
• IPNIT|DIIS_START=ipnit First iteration for DIIS interpolation.
• IPSTEP|DIIS_STEP=ipstep Iteration increment for DIIS interpolation.
• MAXDIS|MAXDIIS=maxdis Max number of Fock matrices used in DIIS interpolation (default 10).
• DIRECT (logical). If given, do integral-direct HF.
• THRMIN|THRDSCF_MIN=value Final integral screening threshold for DSCF.
• THRMAX|THRDSCF_MAX=value Initial integral screening threshold for DSCF.
• THRINT|THRDSCF=value Same as THRDSCF_MIN.
• PRESCREEN=value If nonzero, use density screening (default).
• DISKSIZE]=value Max disk size in Byte for semi-direct calculations (currently disabled).
• BUFSIZE=value Max memory buffer size for semi-direct calculations (currently disabled).
• THRDISK=value Threshold for writing integrals to disk (currently disabled).
• PRINT_DFOCK=value Print option for direct Fock matrix calculation.
• NATORB=record Save natural charge orbitals in given record.
• UNOMIN=unomin Minimum occpation number for UNO-CAS (default 0.02)
• UNOMAX=unomax Maximum occupation number for UNO-CAS (default 1.98)
• POLARI=value If nonzero, compute analytical dipole polarizabilities. See also the POLARI directive (section polarizabilities), which allows to specify various one-electron operators (by default, the dipole operator is used).
• THRCPHF=thresh Threshold for CPHF if polarizabilities are computed (default 1.d-6).
• PRINT|ORBPRINT=value Number of virtual orbitals to be printed. If value=0, the occupied orbitals are printed.
• DEBUG=value Option for debug print.
• THRPRINT=value Threshold for printing orbitals

thrprint=-1 : column-wise;
thrprint=0 : row-wise, as in Molpro2015 and earlier versions ;
thrprint $>0$: print only coefficients that are larger than the threshold together with labels (default: thrprint=0.25)

Since Molpro 2021, it is possible to use the SO-SCI optimization of multi (see so-sci) within the RHF program. All open-shell orbitals are added to the active space and optimized on second-order level. The remaining closed-virtual optimization is approximated on first-order level. This often solves convergence problems for difficult open-shell systems.

The program automatically selects (active) orbitals which are optimized on second-order level. The selection is based on the homo and lumo energies of the starting orbitals, and orbitals with a similar energy are included in the second-order optimization. In the open-shell case, the homo energy of the doubly occupied orbitals is considered. The following options are available for the selection:

• SO_SCI_THRESH Energy threshold for selecting the active orbitals (default: 0.01)
• SO_SCI_MAX_2 Maximum number of closed-shell orbitals treated on second-order level (default: 5)
• SO_SCI_MAX_0 Maximum number of virtual orbitals treated on second-order level (default: 5)

In case of open-shell HF, only the open-shell orbitals are treated as active orbitals as the default, and the selection procedure is skipped. It can be reactivated by setting SO_SCI_THRESH=0.01.

Furthermore, it is possible to perform a second-order HF optimization by adding all closed-shell orbitals to the active space (e.g. SO_SCI_THRESH=10000,SO_SCI_MAX_2=10000,SO_SCI_MAX_0=0).

The number of electrons and the total symmetry of the wavefunction are specified on the WF card:

WF,elec,sym,spin

where

• elec is the number of electrons
• sym is the number of the irreducible representation
• spin defines the spin symmetry, $spin=2*S$ (singlet=0, doublet=1, triplet=2 etc.)

Note that these values take sensible defaults if any or all are not specified (see section symmetry). For example, {rhf; wf,sym=N,spin=M} gives the explicit values N for the symmetry and M for the spin of the wave function, but uses the default number of electrons.

OCC,$n_1,n_2,\ldots,n_8$

To avoid convergence problems in cases with high symmetry, this card should be included whenever the occupation pattern is known in advance. $n_i$ is the number of occupied orbitals in the irreducible representation $i$. The total number of orbitals must be equal to (elec+spin)/2 (see WF card).

CLOSED,$n_1,n_2,\ldots,n_8$

This optional card can be used in open-shell calculations to specify the number of closed-shell orbitals in each symmetry. This makes possible to force specific states in the absence of an OPEN card.

OPEN,$orb_1.sym_1,orb_2.sym_2,\ldots,orb_n.sym_n$

This optional card can be used to specify the singly occupied orbitals. The number of singly occupied orbitals must be equal to spin, and their symmetry product must be equal to sym (see WF card). If the OPEN card is not present, the open shell orbitals are selected automatically. The algorithm tries to find the ground state, but it might happen that a wrong state is obtained if there are several possibilities for distributing the open shell electrons among the available orbitals. This can also be avoided using the CLOSED card. If $orb_i.sym$ is negative, this orbital will be occupied with negative spin (only allowed in UHF).

ORBITAL,record.file
SAVE,record.file

The optimized orbitals, and the corresponding density matrix, fock matrix, and orbital energies are saved on record.file. SAVE is an alias for ORBITAL. If this card is not present, the defaults for record are:

• RHF 2100
• UHF 2200 (holds both $\alpha$ and $\beta$-spin orbitals and related quantities)

These numbers are incremented by one for each subsequent calculation of the same type in the same input. Note that this holds for the sequence number in the input, independently in which order they are executed (see section records).

The default for file is 2.

The START directive can be used to specify the initial orbitals used in the SCF iteration. It is either possible to generate an initial orbital guess, or to start with previously optimized orbitals. Alternatively, one can also use a previous density matrix to construct the first fock operator.

If the START card is absent, the program tries to find suitable starting orbitals as follows:

• First: Try to read orbitals from record specified on the ORBITAL or SAVE card or the corresponding default (see ORBITAL). All files are searched.
• Second: Try to find orbitals from a previous SCF or MCSCF calculation. All files are searched.
• Third: If no orbitals are found, the starting orbitals are generated using approximate atomic densities or eigenvectors of $h$ (see below).

Since these defaults are usually appropriate, the START card is not required in most cases.

An initial orbital guess can be requested as follows:

START,[TYPE=]option

The option keyword can be:

• H0 Use eigenvectors of $h$ (core Hamiltonian) as starting guess.
• ATDEN Use natural orbitals of a diagonal density matrix constructed using atomic orbitals and atomic occupation numbers (default).

Note that it is also possible to use orbitals from previous (e.g., smaller basis set) calculations as starting orbitals (see section starting with previous orbitals below).

Example:

examples/h2o_sto3gstart1.inp
r=1.85,theta=104          !set geometry parameters
geometry={O;              !z-matrix geometry input
H1,O,r;
H2,O,r,H1,theta}
basis=STO-3G              !first basis set
hf                        !scf using STO-3G basis
basis=6-311G              !second basis set
hf                        !scf using 6-311G basis set

The second calculation uses the optimized orbitals of the STO-3G calculation as starting guess. This is done by default and no START card is necessary. The explicit use of START and SAVE cards is demonstrated in the example in the next section.

The following input is entirely equivalent to the one in the previous section:

examples/h2o_sto3gstart2.inp
r=1.85,theta=104          !set geometry parameters
geometry={O;              !z-matrix geometry input
H1,O,r;
H2,O,r,H1,theta}
basis=STO-3G              !first basis set
hf                        !scf using STO-3G basis
start,atdens              !use atomic density guess
save,2100.2               !save orbitals to record 2100.2
basis=6-311G              !second basis set
hf                        !scf using 6-311G basis set
save,2101.2               !save optimized orbitals to record 2101.2

Beware, however, that STO-3G is a very poor basis set and usually gives very bad starting vectors. This technique is best used in combination with reasonable small basis sets (e.g., cc-pVDZ or def2-SVP) or minimal basis sets with a proper repesentation of the occupied atomic orbitals (e.g, 6-31G or MINAO/MINAO-PP):

examples/minao_startorb.inp
memory,50,m

! CuO2(NH3)4
geometry={
N    -2.244097     0.000000    -0.017684
N     1.111652    -1.671116    -0.178974
N     1.111652     1.671116    -0.178974
N    -0.306059     0.000000    -2.311080
Cu    -0.179579     0.000000    -0.290548
O    -0.184327     0.000000     1.594789
O     1.053213     0.000000     2.103145
H    -2.331519     0.000000     1.002818
H     0.614110     0.000000    -2.753879
H    -0.793743     0.815054    -2.687415
H    -0.793743    -0.815054    -2.687415
H    -2.764729     0.814552    -0.345484
H    -2.764729    -0.814552    -0.345484
H     0.635193     2.543632     0.052739
H     0.635193    -2.543632     0.052739
H     1.616225     1.397092     0.672915
H     1.616225    -1.397092     0.672915
H     1.809254     1.910324    -0.883324
H     1.809254    -1.910324    -0.883324
}
wf,charge=1,spin=2

! select all-electron minimal basis sets for H,N,O and ECP10 based basis
! set for Cu; using MINAO-PP here instead of MINAO allows us to project
! the obtained wave function on the cc-pVTZ-PP basis later on.
basis=MINAO,Cu=MINAO-PP

! Run HF to get an initial guess for the valence electronic
! structure. The level shifts damp and stabilize the convergence.
{rhf; shift,-1.0,-0.5; save,2100.2}

! select the actual basis set and start RHF with projected wave function
! from MINAO basis. nitord=1 asks RHF to reorder orbitals in each
! iteration to maximize overlap with the closed and active space
! from the last iteration.
basis=AVTZ,Cu=VTZ-PP,H=VDZ(p)
{df-rhf,nitord=1; start,2100.2}

START,[RECORD=]record.file,[specifications]

reads previously optimized orbitals from record record on file file. Optionally, a specific orbital set can be specified as described in section selecting orbitals and density matrices (ORBITAL, DENSITY).

The specified dump record may correspond to a different geometry, basis set, and/or symmetry than used in the present calculation. Using starting orbitals from a different basis set can be useful if no previous orbitals are available and the ATDENS option cannot be used (see above).

The following example shows how to change the symmetry between scf calculations. Of course, this example is quite useless, but sometimes it might be easier first to obtain a solution in higher symmetry and then convert this to lower symmetry for further calculations.

examples/h2o_c2v_cs_start.inp
r1=1.85,r2=1.85,theta=104          !set geometry parameters
geometry={O;                       !z-matrix geometry input
H1,O,r1;
H2,O,r2,H1,theta}
basis=vdz
{hf                                !scf using c2v symmetry
orbital,2100.2}                    !save on record 2100.2

symmetry,x

{hf
orbital,2101.2}                    !save new orbitals

geometry={O;                       ! geometry has to be respecified so that
H1,O,r1;                 ! H1 and H2 can be retagged as symmetry related
H2,O,r2,H1,theta}
symmetry,x,y
{hf
orbital,2102.2}                    !save new orbitals

Note, however, that this only works well if the orientation of the molecule does not change. Sometimes it might be helpful to use the noorient option.

Note also that a single dump record cannot hold orbitals for different basis dimensions. Using save=2100.2 in the second calculation would therefore produce an error.

If orbitals from a corresponding SCF calculation at a neighbouring geometry are available, these should be used as starting guess.

START,DENSITY=record.file,[specifications]

A density matrix is read from the given dump record and used for constructing the first fock matrix. A specific density matrix can be specified as described in section selecting orbitals and density matrices (ORBITAL, DENSITY). It is normally not recommended to use the DENSITY option.

Using the automated valence active space procedure (AVAS) procedure is often useful to generate a suitable starting guess for open-shell Hartree-Fock calculations. If no previous orbitals are available, AVAS first uses an atomic density guess to create initial orbitals, and these are the reordered according to input specifications. For example, d-orbitals in transition metal complexes can be rotated to the top of the occupied space. For more details see section Automated construction of atomic valence active spaces.

ROTATE,$orb_1.sym,orb_2.sym,angle$

Performs a $2\times 2$ rotation of the initial orbitals $orb_1$ and $orb_2$ in symmetry sym by angle degrees. With angle$=0$ the orbitals are exchanged. See MERGE for other possibilities to manipulate orbitals. In UHF, by default only the $\beta$-spin orbitals are rotated. The initial $\alpha$-spin orbitals can be rotated using

ROTATEA,$orb_1.sym,orb_2.sym,angle$

In this case ROTATEB is an alias for ROTATE.

Since MOLPRO can handle only Abelian point-groups, there may be more symmetry than explicitly used. For instance, if linear molecules are treated in $C_{2v}$ instead of $C_{\infty v}$, the $\delta_{(x^2-y^2)}$-orbitals appear in symmetry 1 ($A_1$). In other cases, a linear geometry may occur as a special case of calculations in $C_S$ symmetry, and then one component of the $\pi$-orbitals occurs in symmetry 1 ($A'$). The program is able to detect such hidden “extra” symmetries by blockings in the one-electron hamiltonian $h$ and the overlap matrix $S$. Within each irreducible representation, an “extra” symmetry number is then assigned to each basis function. These numbers are printed at the end of the integral output. Usually, the extra symmetries are ordered with increasing $l$-quantum number of the basis functions. This information can be used to determine and fix the extra symmetries of the molecular orbitals by means of the SYM command.

SYM,$irrep,sym(1),sym(2),,,sym(n)$

$sym(i)$ are the extra symmetries for the first $n$ orbitals in the irreducible representation irrep. For instance, if you want that in a linear molecule the orbitals 1.1 to 3.1 are $\sigma$ and 4.1, 5.1 $\delta$, the SYM card would read (calculation done with X,Y as symmetry generators):

SYM,1,1,1,1,2,2

If necessary, the program will reorder the orbitals in each iteration to force this occupation. The symmetries of occupied and virtual orbitals may be specified. By default, symmetry contaminations are not removed. If irrep is set negative, however, symmetry contaminations are removed. Note that this may prevent convergence if degenerate orbitals are present.

EXPEC,$oper_1,oper_2,\ldots,oper_n$

Calculates expectation values for one-electron operators $oper_1$, $oper_2$, $\ldots$, $oper_n$. See section One-electron operators and expectation values (GEXPEC) for the available operators. By default, the dipole moments are computed. Normally, it is recommended to use the GEXPEC directive if expectation values for other operators are of interest. See section One-electron operators and expectation values (GEXPEC) for details.

POLARIZABILITY[,$oper_1,oper_2,\ldots,oper_n$]

Calculates polarizabilities for the given operators $oper_1$, $oper_2$, $\ldots$, $oper_n$.. See section One-electron operators and expectation values (GEXPEC) for the available operators. If no operators are specified, the dipole polarizabilities are computed.

Presently, this is working only for closed-shell without direct option.

The polarizabilities are stored in the variables POLXX, POLXY, POLXZ, POLYY, POLYZ, POLZZ.

All commands described in this section are optional. Appropriate default values are normally used.

SHIFT,shifta,shiftb

A level shift of shifta and shiftb hartree for $\alpha$- and $\beta$-spin orbitals, respectively, is applied. This can improve convergence, but has no effect on the solution. shifta$=-0.2$ to $-0.3$ are typical values. The defaults are shifta$=0$ and shifta$=-0.3$ in closed and open-shell calculations, respectively, and shiftb$=0$.

Applying large negative level shifts like {rhf; shift,-1.0,-0.5} will often stabilize convergence at the expense of making it somewhat slower. See section handling difficult cases: When SCF does not converge.

MAXIT,maxit

sets the maximum number of iterations to maxit. The default is maxit$=60$.

ACCU,accu This threshold applies to the square sum of the density matrix element changes (same as option ACCU). If accu$> 1$, a threshold of $10^{-accu}$) is used. The default depends on the global ENERGY threshold, and the threshold is automatically tightened in geometry optimizations or frequency calculations (unless a tight threshold is given).

NOENEST

This disables the sanity check on the energy even if the energy value is unreasonable. Otherwise, the energy will be automatically checked by default.

ORBPRINT,print,test

This determines the number of virtual orbitals printed at the end of the calculation. By default, print$=0$, i.e., only the occupied orbitals are printed. print$=-1$ suppresses printing of orbitals entirely. test$=1$ has the additional effect of printing the orbitals after each iteration.

IPOL,iptyp,ipnit,ipstep,maxdis

This command controls iterative subspace interpolation. iptyp can be:

• DIIS direct inversion of the iterative subspace. This is the default and usually yields fast and stable convergence.
• KAIN Krylov-subspace accelerated inexact newton. A method similar to DIIS.
• NONE No interpolation.

$ipnit$ is the number of the iteration in which the interpolation starts (default: as soon as possible). ipstep is the iteration increment between interpolations (default: 1, i.e., every iteration). maxdis is the maximum dimension of the DIIS matrix (default 10). iptyp and maxdis can also be set as options. E.g.,

{rhf,maxdis=20,iptyp=’DIIS’; shift,-1.0,-0.5}

ORTH,nitort

The orbitals are reorthonormalized after every nitort iterations. The default is nitort$=10$.

DIRECT,options

If this card is present, the calculation is done in direct mode. See section INTEGRAL-DIRECT CALCULATIONS (GDIRECT) for options. Normally, it is recommended to use the global GDIRECT command to request the direct mode. See section INTEGRAL-DIRECT CALCULATIONS (GDIRECT) for details.

General suggestions:

• Carry out convergence experiments with a small but reasonable basis set (e.g., cc-pVDZ, def2-SVP, aug-cc-pVDZ, ASVP). STO-3G is not a reasonable basis set.

Before you start you should check:

• Whether your geometry is sensible (e.g., look for Angstrom/Bohr conversion issues). Note that Molpro prints bond distances in both Angstroms and atomic units at the top of an output.
• Whether you have selected the correct electronic state (spin and symmetry). Molpro tries to guess spatial symmetries of open-shell compounds automatically if none are provided. However, the guess is not always right. In such a case you need to give the symmetry manually (in the simplest case as wf,sym=N,spin=M. See section defining the wavefunction). Molpro does not attempt to guess the spin state of the input compound automatically; it defaults to spin=0 for systems with even numbers of electrons and spin=1 for odd-numbered species.

If convergence problems persist, it is recommended to use the new mixed first-order/second-order MCSCF program (MULTI,SO-SCI, or (recommended for larger systems) DF-MULTI (SO-SCI is default for DF-MULTI) as described in D.A. Kreplin, P.J. Knowles and H.-J. Werner, J. Chem. Phys. 152, 074102 (2020); https://doi.org/10.1063/1.5142241. This provides very robust convergence, even in cases where RHF does not converge or converges to a wrong solution. Due to faster convergence and only small additional cost per iteration the overall computation time is often even smaller than for conventional RHF or DF-RHF. For MULTI it is necessary to specify the configuration uniquely using OCC, CLOSED, and WF directives. See section The MCSCF program MULTI for details.

With older Molpro versions, which do not contain the new MCSCF program, the following techniques can be attempted:

Hartree-Fock options & Small-basis initial guess:

• Level shifts: Adding a level shift like {rhf; shift,-1.0,-0.5} stabilizes the current RHF solution against changes and leads to smoother (but slower) convergence. That should be your first try; it is often sufficient.
• Occupation freezing: The option {rhf,nitord=N} can be used to freeze orbital occupations at iteration N. When the programs emits warnings about reassigned orbital occupations, you could try to freeze the occupations only later (give a higher N) or earlier (give a smaller N). By freezing the occupation pattern you tell the RHF program to try to lock on whatever solution it currently is pursuing. This often helps if multiple RHF solutions with similar energies are present and otherwise the program would oscillate between some of them. Note that {rhf,nitord=1} will tell RHF to lock onto the initial occupation; if combined with orbital rotation or advanced initial guesses this can often be used to converge to specific solutions (e.g., some excited states).
• Minimal-basis SCF guess: Try to obtain a Hartree-Fock solution with a minimal-basis AO set first and to use this as initial guess for the actual Hartree-Fock calculation. For this purpose we provided the basis set definition “MINAO” (to complement cc-pVnZ basis sets) and “MINAO-PP” (to complement cc-pVnZ-PP sets with ECPs). See section initial orbital guess for an example. These sets simply consist of the AO part of the cc-pVTZ or cc-pVTZ-PP basis sets, stripped of all their polarization functions. Since a minimal basis has fewer degrees of freedom than a real basis set, convergence is often easier, and it can still provide reasonable guess for the valence electronic structure. Note: The MINAO basis sets are very small, so conventional Hartree-Fock (in integral-direct mode if necessary) is typically much faster than density-fitting Hartree-Fock.
• Increasing the DIIS dimension: In rare cases {rhf,maxdis=30,iptyp=’DIIS’,nitord=20; shift,-1.0,-0.5} can find solutions which are not found in the standard settings. Usually increasing the DIIS dimension beyond 10 (the default) just slows down convergence. It is also worthwile to try a variation of DIIS known as KAIN (Krylov-subspace accelerated inexact Newton): {rhf,maxdis=10,iptyp=’KAIN’,nitord=10; shift,-1.0,-0.5} which sometimes shows different convergence behavior than straight DIIS.

Cationic or Anionic initial guess:

• If molecule $X$ does not converge, it might still be possible to converge $X^+$ ($X^{2+}$, ..) and use this as initial guess for the actual computation. Particularly if $X^+$ is a closed-shell compound this will often work. If using this technique, you need to be careful about the final state you arrive in. Because your initial guess is biased, the $X$ calculation might converge to an excited state.

Density-functional initial guess:

• For transition metals and transition states sometimes DFT methods show better convergence behavior than RHF. You might perform a DFT calculation (possibly with a smaller basis set) and use it as initial guess for the SCF. E.g., {df-rks,b-lyp; coarsegrid; save,2100.2} {df-rhf,nitord=1; orbital,2100.2}

If all else fails: Use the MCSCF program. The MCSCF program uses an advanced orbital optimization algorithm which is much more robust than the SCF method, and which can converge almost everything you give to it (but it is often slower and sometimes locks on an excited state if started from an atomic density guess). MCSCF can also calculate Hartree-Fock solutions if used with suitable input cards.

It is possible to calculate the eigenvalues of the orbital Hessian with the MCSCF program (see orbital Hessian eigenvalues ) for a RHF wavefuction. This requires a manually given orbital occupation in the MCSCF program (see CLOSED and OCC) similar to the RHF solution. The computation of the orbital Hessian is considerably more expansive than the RHF optimization. Negative eigenvalues show that the solution is a saddle point, and a lower energy could be found.

Some kinds of excited or higher ionized states can be calculated using a so-called delta-SCF procedure. Here the excitation/ionization energy is obtained as difference of the SCF solutions for the ground state and the excited/ionized state. If the excited state is reasonably well described by a single determinant, this procedure is normally quite accurate.

The example (requires pyridine.xyz geometry file)

examples/pyridine-N1s-core-hole.inp
! This example calculates the N 1s core-hole binding energy of pyridine
! using a delta-SCF procedure. It calculates a SCF solution in the
! ground state, then in the core-excited state, and reports the
! difference. The example shows how to obtain the core-excited state
! of a core of your choice.
geometry=pyridine.xyz

! we use default basis sets for all atoms except the one we wish
! to calculate the core hole of, in this case N 1s. For this atom,
! we uncontract the s and p functions of the basis to give the core
! hole more opportunities for relaxation.
!
! Notes:
!   - Uncontracting the complete basis would of course also work,
!     but might be more expensive
!   - For correlated calculations, you would need additional tight
!     correlation functions for the hole (e.g., use a cc-pwCVnZ basis
!     on the affected atom and cc-pVTZ on the rest)
basis={
default,def2-TZVPP
sp,N,def2-TZVPP     ! no "c;" following -- uncontracted.
df,N,def2-TZVPP;c;  ! use d and f functions with contractions.
}

! reference calculation, normal DFT. We also uncontract the fitting
! basis to fit the core region more accurately.
{df-rks,pbe,df_basis=def2-tzvpp(u)}
ENORMAL = ENERGY       ! remember energy of ground state calculation.

! localize orbitals. This isolates the cores of the individual atoms
! also in the case of degeneracy. And, in particular, it shows us which
! orbital is the N 1s we are looking for. It comes out as orbital 1.1.
{ibba; save,2101.2}

! do the core-hole calculation. Additionally to the previous command, we
! specify nitord=1, which asks SCF to fix orbital order & occupations
! starting at iteration 1. This should prevent SCF from dropping down to
! the ground state SCF solution.
!
! With this, SCF will emit warnings about strongly deviating orbital
! occupations in the first iterations. This is to be expected when
! treating instable states and no reason for concern.
{df-rks,pbe,df_basis=def2-tzvpp(u),nitord=1;
! in the program, closed-shell orbitals always have lower numbers
! than open-shell orbitals. In order to get the core hole state, we
! thus need to exchange the localized N1s orbital (1.1) with whatever
! came out at the highest orbital number (21.1) before. [Because the
! latter one will be considered as singly-occupied. If we would not use
! localized orbitals, 21.1 would be the HOMO].
rotate,1.1,21.1;    ! exchange orbitals 1.1 and 21.1
orbital,2101.2;     ! use the localized orbitals as input
wf,spin=1,charge=1  ! one open-shell orbital (implicitly 21.1)
}
EWITH_HOLE = ENERGY
! run population analysis again to see if we arrived at the right
! state. This should be indicated by an active orbital occupation
! of about 1.0 on the N 1s orbital.
{ibba}
! show the result, converted to electron volts.
{table,(EWITH_HOLE-ENORMAL)*toev
title,N 1s core hole binding energy [Exp. value: (404.94 +/- 0.03) eV. [Can J. Chem 58 694 (1980)]]
}

shows how to use the delta-SCF procedure to calculate the N 1s core binding energy in pyridine, by employing localized orbitals and moving them to the HOMO position using a ROTATE card (Sec. rotating pairs of orbitals). In the core-excited state calculation, SCF is prevented from collapsing to the ground state by using the option NITORD=1 (Sec. options), which asks SCF to lock onto the input solution.