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Subsections
6.3 caspt2
Second order multiconfigurational perturbation theory is used in the program
CASPT2 [15,16] to compute the (dynamic)
correlation energy. The reference state is
usually of the CAS type, but the program has been extended to also accept
RAS reference states[17,18].
The first step is therefore a RASSCF calculation and the CASPT2
calculation gives a second order estimate of the difference between the RASSCF
and the full CI energy. For calculations using a true RAS reference, benchmark
calculations were reported by Sauri et al. [18]. For CASSCF
references, the CASPT2 method has been tested in a large number
of applications [19,20]. Here follows a brief summary of
results.
Bond distances are normally obtained with an accuracy of better that 0.01
Å for bonds between first and second row atoms. With the standard Fock
matrix formulation, bond energies are normally underestimated with between 2
and 5 kcal/mol for each bond formed. This is due to a systematic error in the
method[21]. In every process where the number of paired
electrons is changed, an error of this size will occur for each electron pair.
For example, the singlettriplet energy difference in the methylene radical
(CH_{2}) is overestimated with about 3 kcal/mol [16]. Heats of
reactions for isogyric reactions are predicted with an accuracy of
kcal/mol. These results have been obtained with saturated basis sets and all
valence electrons active. The use of smaller basis sets and other types of
active spaces may, of course, affect the error.
These systematic errors have recently been considerably reduced by the
introduction of a modified zeroth order Hamiltonian [22]. The method
introduces a shift (the IPEA shift) that modifies the energies of active
orbitals such that they become closer to ionization energies when excited from
and closer to electron affinities when excited out of. The approach has been
tested for 49 diatomic molecules, reducing the mean error in D_{0} from 0.2 to
0.1 eV. For the triply bonded molecules N_{2}, P_{2}, and As_{2} it was reduced
from 0.45 eV to less than 0.15 eV. Similar improvements were obtained for
excitation and ionization energies. The IPEA modified H_{0} (with a shift
parameter of 0.25) is default in MOLCAS from version 6.4.
An alternative to IPEA is to use the options, called `g_{1}', `g_{2}', and
`g_{3}'(See Ref. [23]), that stabilizes the energies of the
active orbitals. The remaining error is no longer systematic, and is generally
reduced. For example, the error in the singlettriplet separation of CH_{2} is
reduced to 1 kcal/mol [23]. This option is, however, not
recommended any longer because it has been replaced by the IPEA Hamiltonian.
The CASPT2 method can be used in any case where a valid reference function can
be obtained with the CASSCF method. There is thus no restriction in the number
of open shells or the spin coupling of the electrons. Excited states can be
treated at the same level as ground states. Actually one of the major
successes with the method has been in the calculation of excitation energies.
A large number of applications have been performed for conjugated organic
molecules. Both Rydberg and valence excited states can be treated and the
error in computed excitation energies is normally in the range 0.00.2 eV.
Similar results have been obtained for ligand field and chargetransfer
excitations in transition metal compounds. From MOLCAS6 it is possible to use
the CASPT2 method in conjunction with the DouglasKrollHess relativistic
Hamiltonian, which has made possible calculations on heavy element compounds
such a third row transition metal compounds and actinides with accurate results.
The CASPT2 method can also be used in combination with the FFPT
program to compute dynamic correlation contributions to properties with good
results in most cases. Numerical gradients are available with the
slapaf module.
The CASPT2 method is based on second order perturbation theory. To be
successful, the perturbation should be small. A correct selection of
the active space in the preceding CASSCF calculation is therefore of
utmost importance. All neardegeneracy effects leading to configurations
with large weights must be included at this stage of the calculation.
If this is not done, the first order wave function will contain large
coefficients. When this occurs, the CASPT2 program issues a warning.
If the energy contribution from such a configuration is large, the results is
not to be trusted and a new selection of the active space should be made.
Especially in calculations on excited states, intruder states may occur in the
first order wave function. Warnings are then issued by
the program that an energy denominator is small or negative. Such intruder
states often arise from Rydberg orbitals, which have not been included in the
active space. Even if this sometimes leads to large first order CI coefficients,
the contribution to the second order energy is usually very small, since the
interaction with the intruding Rydberg state is small. It might then be
safe to neglect the warning. A safer procedure is to include the Rydberg
orbital into the active space. It can sometimes be deleted from the MO space.
Calculations on compounds with heavy atoms (transition metals, actinides, etc)
may yield many virtual orbitals with low energies. The interaction energies for
excitations to states where these orbitals are occupied are often very small and
the low denominators can then be removed by a suitable level shift (see below).
But it is always safer to include such orbitals in the active space.
Two keywords have been introduced to deal with this fairly common
situation, for excited states, that weakly coupled intruders cause
spurious singularities, `spikes' in e.g. a potential curve. The two
keywords SHIFT and IMAGINARY SHIFT (mutually exclusive) will introduce a shift
in the energy denominators,
thus avoiding singularities, and will also correct the energy for the use of
this shift. The net effect is that the energy is almost unaffected except in
the vicinity of the weak singularity, which is removed. The SHIFT keyword adds
a real shift, and the use of this procedure is well tested
[24,25]. The IMAGINARY SHIFT adds an imaginary quantity, and
then uses the real value of the resulting secondorder energy
[26]. This offers some advantage, in particular for weak intruder
states.
In some cases, where one can expect strong interaction between different CASSCF
wave functions, it is advisable to use the MultiState (MS) CASPT2 method
[11]. A second order effective Hamiltonian is constructed for a
number of CASSCF wave functions obtained in a stateaverage calculation. This
introduces interaction matrix elements at second order between the different
CASSCF states. The effective Hamiltonian is diagonalized to obtain the final
second order energies. The program also produces a file, JOBMIX, with the new
effective zeroth order wave functions, which are linear combinations of the
original CASSCF states. This method has been used successfully to separate
artificially mixed valence and Rydberg states and for transition metal compounds
with low lying excited states of the same symmetry as the ground state.
It is clear from the discussion above that it is not a `black box' procedure
to perform CASPT2 calculations on excited states. It is often necessary to
iterate the procedure with modifications of the active space and the selection
of roots in the CASSCF calculation until a stable result is obtained. Normally,
the CASSCF calculations are performed as average calculations over the number
of electronic states of interest, or a larger number of states.
It is imperative that the result is checked
before the CASPT2 calculations are performed. The solutions should contain
the interesting states. If all of them are not there, the number of roots in
the CASSCF calculation has to be increased. Suppose for example, that four
states of a given symmetry are required. Two of them are valence excited states
and two are Rydberg states. A CASSCF calculation is performed as an average
over four roots. Inspection of the solution shows only one valence excited
state, the other three are Rydberg states. After several trials it turns out
that the second valence excited state occurs as root number seven in the
CASSCF calculation. The reason for such a behavior is, of course, the
very different dynamic correlation energies of the valence excited states as
compared to the Rydberg states. It is important that the AO basis set is
chosen to contain a good representation of the Rydberg orbitals, in order to
separate them from the valence excited states. For more details on how to
perform calculations on excited states we refer to the
literature [24,25] and section of
the examples manual.
The first order wave function is obtained in the CASPT2 program as an
iterative solution to a large set of linear equations. The size of the
equation system is approximately n^{2}*m^{2}/2 where n is the sum of inactive
and active orbitals and m is the sum of active and secondary orbitals.
Symmetry will reduce the size with approximately a factor , the
number of irreps of the point group.
CASPT2 produces a set of molecular orbitals that can be used
as start orbitals for other programs or further calculations.
A minimal CASSCF and CASPT2 gives orbitals and occupation numbers
which can be used to design a proper larger calculation.
By default, the orbitals are natural orbitals obtained from the
density matrix of the (normalized) wave function through first order.
However, the active/active block of that density matrix is not computed
exactly. An approximation has been designed in such a way that the trace
is correct, and the natural occupation numbers of active orbitals are
between zero and two. Due to the approximation, any properties computed
using these orbitals are inexact and can be used only qualitatively. An exact
first order density matrix can be computed but this is more timeconsuming. It
is controlled by the keyword DENSity. Use this keyword to compute
properties like dipole moments, etc. The most secure accurate way to do that is.
however, to use finite field perturbation theory (FFPT).
For compatibility with earlier programs, two keywords are available
that change the default definition of the output orbitals.
Using the keyword MOLOrb, you will obtain orbitals that are
identical to the natural orbitals from the RASSCF calculation
in the inactive and active subspaces, while the secondary orbitals are
obtained by diagonalizing the secondary subspace of the density matrix of the
(normalized) perturbed wave function. This is often useful for preparing
orbital sets for subsequent calculations. The RASSCF calculation
can be reproduced with any or several virtual orbitals deleted. Therefore,
the virtual space can be trimmed by deleting
orbitals with low occupation number.
Also, an intruder due to a deficient
active space will produce a virtual orbital with large occupation number.
Inclusion of this orbital into the active space eliminates the intruder.
Similarly, if the intruder is of the weak `accidental' type, that orbital
can be deleted.
Using the NATUral keyword, you will get the natural orbitals obtained from
the density matrix through first order, either in the approximate form (default)
or in the exact form by the use of the keyword DENSity.
6.3.1 Dependencies
The CASPT2 program needs the JOBIPH file from a RASSCF
calculation, and in addition one and twoelectron integrals and some auxiliary
files from SEWARD.
6.3.2 Files
CASPT2 will use the following input
files: ONEINT, ORDINT,RUNFILE, JOBIPH
(for more information see ).
File  Contents

PT2ORB  Molecular orbitals.

6.3.3 Input
This section describes the input to the CASPT2 program, starting with its name:
&CASPT2
Keyword  Meaning

TITLe  This keyword is followed by one title line.

MULTistate  Enter number of root states, and a list of which CI vector from
the CASSCF calculation to use for each state, for example ``2 1 2 ''
would specify the first and second root.
Also used for singlestate calculations, when the root state is not
the ground state, for example ``1 2 '' would specify the second root.

IPEAshift  This shift corrects the energies of the active orbitals and is
specified in atomic units. It will be weighted by a function of the
diagonal density matrix element .
This option is used to modify the standard definition of the
zeroth order Hamiltonian (H_{0}), which includes an IPEA shift of 0.25
[22]. The modification of H_{0} has been introduced (Nov 2005) to
reduce the systematic error which leads to a relative overestimation of the
correlation energy for open shell system. It also reduces the intruder problems.
Default is to use an IPEA shift of 0.25.

IMAGinary  Add an imaginary shift to the external part of the zero order
Hamiltonian. The correlation energy computed is the real part
of the resulting complex perturbation energy.
Also, a corrected
value, obtained by Hylleraas' variational formula, is computed.
See Ref. [26].
As with the real shift, this option is used to eliminate intruder
problems.

SHIFt  Add a shift to the external part of the zero order Hamiltonian.
See Refs. [26,24,20].
In addition to the conventionally computed second order energy
value, another energy obtained by Hylleraas' variational formula
is computed. This energy is then very close to the unshifted
energy, except close to singularities due to intruders.
This option should only be used to eliminate intruder state problems.

AFREeze  This keyword is used to select atoms for defining the correlation orbital
space for the CASPT2 calculation. Assume that you have a large molecule where
the activity takes place in a limited region (the active site). It could be a
metal atom with its surrounding ligands. You can then use this option to reduce
the size of the CASPT2 calculation by freezing and deleting orbitals that have
only a small population in the active site. An example: The cobalt imido complex
Co^{III}(nacnac)(NPh) has 43 atoms. The active site was cobalt and the
surrounding ligand atoms. Using the AFRE option reduces the time for the CASPT2
calculation from 3 hrs to 3 min with a loss of accuracy in relative energies for
24 electronic states of less than 0.1 eV. The first line after the keyword
contains the number of selected atoms then the selection thresholds (the
recommended value is 0.1 or less). An additional line gives the names of the
atoms as defined in the Seward input. Here is a sample input for the cobalt
complex mentioned above.
AFREeze
6 0.10 0.00
Co N1 N2 C5 C6 C7
This input means that inactive orbitals with less than 0.1 of the density on
the active sites will be frozen, while no virtual orbitals will be deleted.

LOVCaspt2  ``FreezeandDelete'' type of CASPT2, available only in connection with Cholesky or RI.
Needs (pseudo)canonical orbitals from RASSCF. An example of input for the keyword LOVC is the following:
LovCASPT2
0.3
DoMP2 (or DoEnv)
In this case, both occupied and virtual orbitals (localized by the program) are divided in two groups: those mainly located on
the region determined (automatically) by the spatial extent of the active orbitals (``active site''),
and the remaining ones, which are obviously ``outside'' this region.
The value of the threshold (between 0 and 1) is used to perform this selection
(in the example, 30% of the gross Mulliken population of a given orbital on the active site).
By default, the CASPT2 calculation is performed only for the correlating orbitals associated with the active site.
The keyword DoMP2 is optional and forces the program to perform also an MP2 calculation on
the ``frozen region''.
Alternatively, one can specify the keyword VirAll in order to use all virtual orbitals as correlating space for the
occupied orbitals of the active site.
A third possibility is to use the keyword DoEnv to compute the energy of the environment as total MP2 energy
minus the MP2 energy of the active site.

FNOCaspt2  Performs a Frozen Natural Orbital (FNO) CASPT2 calculation, available only in combination with Cholesky or RI integral representation.
Needs (pseudo)canonical orbitals from RASSCF. An example of input for the keyword FNOC is the following:
FNOCaspt2
0.4
DoMP2
The keyword FNOC has one compulsory argument (real number in ]0,1]) specifying the fraction of virtual orbitals
(in each irrep) to be retained in the FNOCASPT2 calculation.
The keyword DoMP2 is optional and used to compute the (estimated) correction for the truncation error.

FOCKtype  Use an alternative Fock matrix. The default Fock matrix is described in
[15,16] and the other original CASPT2 references.
The three different modifications named G1, G2 and G3 are described in
[23].
Note: from 6.4 it is not recommended to use this keyword but
stay with the IPEA modified H_{0}, which is default.

FROZen  This keyword is used to specify the number of frozen orbitals,
i.e. the orbitals that are not correlated in the calculation.
The next line contain the number of frozen orbitals per
symmetry. The default is to freeze the max of those that were frozen in the
RASSCF calculation and the deep core orbitals.
The frozen orbitals are always the first ones in each symmetry.

DELEted  This keyword is used to specify the number of deleted orbitals,
i.e. the orbitals that are not used as correlating orbitals in
the calculation. The next line contain the number deleted orbitals per symmetry.
The default is to delete those that were deleted in the RASSCF
calculation.
The deleted orbitals are always the last ones in each symmetry.

DENSity  Computes the full density matrix from the first order wave function,
rather than approximated as is the (faster) default option. Used to
compute CASPT2 properties, such as dipole moments, etc.

RFPErt  This keyword makes the program add reaction field effects to the energy
calculation. This is done by adding the reaction field effects to the
oneelectron Hamiltonian as a constant perturbation, i.e. the reaction field
effect is not treated self consistently. The perturbation is extracted from RUNOLD,
if that file is not present if defaults to RUNFILE.

RLXRoot  Specifies which root to be relaxed in a geometry optimization of a
multi state CASPT2 wave function. Defaults to the highest root or
root defined by the same keyword in the RASSCF module.

THREsholds  On next line, enter two
thresholds: for removal of zeronorm components in the
firstorder perturbed wave function, and for removal of near linear
dependencies in the firstorder perturbed wave function. Default
values are 1.0d10 and 1.0d08 respectively.

MAXIter  On next line, enter the maximum allowed number of iterations
in a procedure for solving a system of
linear equations using a conjugate gradient method. Default is 20.
A gradient norm is reported. This gradient is a residual error from the
CASPT2 equation solution and should be small, else the number of iterations
must be increased.

CONVergence  On next line, enter the convergence threshold for the procedure described above.
The iterative procedure is repeated until the norm of the residual
(RNORM) is less than this convergence threshold. Default is 1.0d06.

MOLOrb  This keyword gives backwards compatibility to earlier CASPT2 programs.
It specifies that a set of output orbitals will be created, which is
identical to the CASSCF orbitals, except that the virtual orbitals
are the natural orbitals of the (normalized) virtual/virtual
part of the density matrix of the perturbed wave function.

NATUral  This keyword gives backwards compatibility to earlier CASPT2 programs.
It specifies that a set of output orbitals will be created, that
are the natural orbitals of the firstorder density matrix.
Note that it is necessary to use the keyword
DENSity to obtain an exact density matrix. Otherwise, only an
approximate density matrix is created.

NOMIx  Normally, a MultiState CASPT2 calculation produces new jobiph file named
JOBMIX. It has the same CASSCF wave functions as the original ones, except
that those CI vectors that was used in the MultiState CASPT2 calculation
have been mixed, using the eigenvectors of the effective Hamiltonian matrix
as transformation coefficients.
Keyword NOMIX prevents creation of this JOBMIX file.

NOMUlt  This keyword removes the multistate part of the calculation and only runs a
series of independent CASPT2 calculations for the roots specified by the
MULTistate keyword. Useful when many roots are required, but multistate is
not needed, or desired. Note that a JOBMIX file is produced anyway, but the
vectors will not be mixed, and the energies will be singlestate CASPT2
energies.

NOORbitals  In calculations with very many orbitals, use this keyword to skip the
printing of the MO orbitals.

NOPRop  Normally, a CASPT2 calculation produces an exact or approximate density matrix,
natural orbitals, and properties.
Keyword NOPROP inhibits these calculations, saving time and memory.

NOTRansform  This keyword specifies that the wave function should not be transformed
to use quasicanonical orbitals, even if CASPT2 does not know if this
was done or not and by default would do such a transformation.
Effectively, the Fock matrix is replaced by a diagonal
approximation in the input orbital system.

TRANsform  This keyword specifies that the wave function should be transformed
to use pseudocanonical orbitals, even if this was specified
as option to the CASSCF calculation and should be unnecessary.
(Default is: to transform when necessary, and not else.)

OFEMbedding  Adds an OrbitalFree Embedding potential to the hamiltonian. Available only in combination with Cholesky or RI integral representation.
No arguments required. The runfile of the environment subsystem (AUXRFIL) must be available.

GHOStdelete  Excludes from PT2 treatment orbitals localized on ghost atoms. A threshold for this selection must be specified.

OUTPut  Use this keyword, followed by any of the words BRIEF, DEFAULT, or LONG, to
control the extent of orbital listing.
BRIEF gives a very short orbital listing,
DEFAULT a normal output, and LONG a detailed listing.

PRWF  This keyword is used to specify the threshold for printing the
CI coefficients, default is 0.05.

The given default values for the keywords
Convergence and
Thresholds normally give a second order energy which is correct
in eight decimal places.
&CASPT2
Title
The water molecule
Density matrix
The CASPT2 energy and density matrix is computed for the water molecule with the
O(1s) orbital frozen. The standard IPEAH_{0} is used.
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