3.1. Wavefunction Types: RHF/RKS, UHF/UKS, ROKS and more¶
The basic variable that controls the type of wavefunction to be computed
is the variable HFTyp in the %scf block. If nothing is specified for
HFTyp, the program will check the multiplicity given in the input: for
closed-shell molecules with multiplicity 1, RHF/RKS is assumed; for open
shell molecules with multiplicity larger than 1, UHF/UKS is invoked.
RHF will lead to a spin restricted closed-shell type computation
[180] whereas UHF does a spin unrestricted calculation.
For DFT calculations, RKS, UKS and ROKS can be
used as synonyms for RHF, UHF and ROHF. The restricted open-shell
DFT method (ROKS) is only operative for high-spin states that have \(n\)
unpaired electrons and \(S=n/2\). UKS wavefunctions will not be
spin-purified.
%scf
HFTyp RHF # closed-shell (RKS for DFT)
UHF # unrestricted open-shell (UKS for DFT)
ROHF # restricted open-shell (ROKS for DFT)
CASSCF # complete active space SCF
end
In certain cases you may want to run open-shell systems with RHF/RKS to get a “half-electron” type wavefunction [181]. The total energy is not corrected! Sometimes these half-electron computations lead to acceptable convergence, and the resulting orbitals may be used as input for ROHF, UHF or MRCI calculations. Special attention should be taken for transition metal complexes where the RHF/RKS orbitals are quite different from ROHF or UHF orbitals, so that it is not recommended to over-interpret the wavefunctions from such calculations. The calculation is set up in the following way:
%method AllowRHF true end
# or simply: ! AllowRHF
3.1.1. ROHF Options¶
For ROHF calculations[6, 7, 182, 183, 184, 185, 186, 187, 188] the program will try to figure out what type of open-shell situation is present on the basis of the initial guess orbitals and their energies. Most “simple” cases are well recognized, but sometimes a little help from the user is needed.
The simplest ROHF case is the HIGHSPIN case, where all unpaired electrons
in the open-shell are ferromagnetically coupled (parallel to each other), resulting in the highest
multiplicity possible. The user can request this case as follow:
%scf
HFTyp ROHF
ROHF_case HIGHSPIN
ROHF_NEl[1] 4 # number of electrons in the open-shell
end
The ROHF code also has a very powerful feature that goes
back to insights of Mike Zerner [189, 190]. It
can average over either all states of a given configuration (CAHF)
or all states of a given spin for a given configuration (SAHF).
Especially, the SAHF feature gives you easy access to most degenerate
high symmetry situations and the orbitals resulting from such
calculations will be very convenient as input for CI calculations.
%scf
HFTyp ROHF
ROHF_case CAHF # configuration averaged HF
SAHF # spin averaged HF
ROHF_NumOp 3 # number of operators (3, 2 or 1)
ROHF_NOrb[1] 2,1 # number of orbitals in each open-shell
ROHF_NEl[1] 1,1 # number of electrons in each open-shell
end
The hypothetical example below could represent an excited state of an octahedral d\(^{3}\) transition metal complex. In this case there are five open-shell orbitals. The first three open-shell orbitals contain two electrons and the last two one electron. The input for a SAHF calculation is identical, just replace CAHF with SAHF.
%scf
HFTyp ROHF
ROHF_case CAHF # configuration averaged HF
ROHF_NumOp 3 # 3 operators in this case: closed, open1, open2
ROHF_NOrb[1] 3,2 # 3 orbitals in first open shell, 2 in the second
ROHF_NEl[1] 2,1 # 2 electrons in first open shell, 1 in the second
end
Another feature of the ROHF code is the ability to converge the SCF to
a given Configuration State Function (CSF-ROHF) [191].
In this way one can approach results from MCSCF calculations. This can be
requested in two ways.
The user can give a specific coupling situation.
%scf
HFTyp ROHF
ROHF_CASE USER_CSF # User defined CSF
ROHF_REF {1 1 -1 -1} end # CSF to be converged to a 2 open shell antiferromagnetically coupled
end
Or the user can give how many orbitals per shell. Where each open-shell will couple with antiparallel spin with the previous one.
%scf
HFTyp ROHF
ROHF_CASE AF_CSF # User defined CSF
ROHF_AFORBS 2,2 # Coupling Situation
end
As an example, one can think of a Fe(III) dimer, where each center is locally high spin, but they couple antiferromagnetically to each other. In order to get the ROHF solution for this system, first one needs a set of initial guess orbitals. The guess orbitals can be obtained either from the QROs of an UHF calculation, or a high spin ROHF calculation, or even a SAHF or CAHF. Independently of the method used, the orbitals need to be localized and ordered in a way that the set of 3d orbitals of each iron are grouped together in sequence. From this, one can run a CSF-ROHF calculation for the antiferromagnetic CSF as shown bellow:
%scf
HFTyp ROHF
ROHF_CASE USER_CSF # User defined CSF
ROHF_REF {1 1 1 1 1 -1 -1 -1 -1 -1} end # 2 open shells coupling antiparallel
end
or
%scf
HFTyp ROHF
ROHF_CASE AF_CSF # User defined CSF
ROHF_AFORBS 5,5 # 2 open shells coupling antiparallel
end
The CSF-ROHF procedure can recognize doubly occupied and virtual orbitals in the definition of
the CSF when the USER_CSF case is invoked. When detected, these orbitals will be rotated out of
the open-shell defined in the ROHF method and the calculation will run normally:
%scf
HFTyp ROHF
ROHF_CASE USER_CSF # User defined CSF
ROHF_REF {1 1 2 1 -1 -1 0 -1} end # the DOMO will be rotated to the closed-shell and
# the VMO will be rotated to the virtual space.
end
The user can also directly input the ROHF variables by means of the
ROHFOPT Case User keyword. For example,
for the high spin case with three electrons in three orbitals, it gives two
operators with vector coupling coefficients \(a=1\) and \(b=2\) (Zerner
convention).
%scf
HFTyp ROHF
ROHFOP Case User # manual input of ROHF variables
Nop 2 # number of operators
Norb[1] 3 # number of open-shell orbitals
Nel[1] 3 # number of open-shell electrons
A[1,1] 1 # Coulomb vector in the open shell
B[1,1] 2 # Exchange vector in the open shell
end
end
One awkward feature of the ROHF theory is that the Fock operator is somewhat arbitrarily defined. Different choices lead to the same wavefunction, but have different convergence properties that may vary from system to system. ORCA thus lets the user choose the desired variant. Playing around with these choices may turn a divergent or slowly converging ROHF calculation into a successful calculation!
The ROHF_Restrict feature is another feature that may be useful. If
you suspect that the ROHF calculation does not converge because an
open-shell and a closed-shell orbital are flipping back and forth, you
can try to avoid this behavior by choosing ROHF_Restrict true. Of
course there is no guarantee that it will work, and no guarantee that
the system stays in the desired state. However, it decreases the chances
of large, uncontrolled steps.
%scf
ROHF_Mode 0 # construct F according to Pulay (default)
1 # construct F as in the Gamess program
2 # construct F according to Kollmar
ROHF_Restrict false # restrict orbital interchanges and off-diagonal elements
# (default=false)
# a complete list of ROHF variables
ROHFOP
Case User # manual input of ROHF variables
Nop 2 # number of operators
Norb[1] 3 # number of open-shell orbitals
Nel[1] 3 # number of open-shell electrons
A[1,1] 1 # Coulomb vector in the open shell
B[1,1] 2 # Exchange vector in the open shell
Mode 2 # use the Kollmar operator
Restrict false # do not restrict
end
end
3.1.2. UHF Natural Orbitals¶
The program can produce the UHF natural orbitals (UNOs). With these, the open-shell wavefunction can be pictured conveniently. The syntax is simple:
%scf
UHFNO true
end
# or simply: ! UNO
There are various printing options for UNOs described in the output
section Population Analysis. The UNOs can also be plotted as described
in the plots section
Orbital and Density Plots. In general the program stores a file
BaseName.uno, where BaseName is by default the name of your input
file with .inp stripped off. Accordingly, the gbw file is named
BaseName.gbw. The .uno file is a normal gbw file that contains the
geometry, basis set and the UNO orbitals. It could be used, for example,
to start a ROHF calculation.
3.2. UNO, UCO and QROs input¶
For calculations on open-shell systems we recommend to use the keywords
!UNO !UCO in the input line. This will generate quasi-restricted
molecular orbitals QRO, unrestricted natural spin-orbitals UNSO,
unrestricted natural orbitals UNO and unrestricted corresponding
orbitals UCO. Moreover, it will print the UCO overlaps in the output,
which can provide very clear information about the spin-coupling in the
system. Below an example of the input and section of the output is
provided.
!B3LYP def2-SVP UNO UCO TightSCF
The UCO overlap section in the output will look like:
***UHF Corresponding Orbitals were saved in MyJob.uco***
----------------------
Orbital Overlap(*)
----------------------
.
.
.
96: 0.99968
97: 0.99955
98: 0.99947
99: 0.99910
100: 0.99873
101: 0.99563
102: 0.74329
103: 0.00000
The overlap that corresponds to a value usually less than 0.85 denotes a spin-coupled pair. Whereas, values close to 1.00 and 0.00 refers to the doubly occupied and singly occupied orbitals respectively.