5.24. Electron Paramagnetic Resonance (EPR) Parameters¶
Electron paramagnetic resonance probes the magnetic sublevels of the
electronic ground state. In the %eprnmr
input block, calculations of
the parameters of a spin Hamiltonian of the form
can be requested. The properties are evaluated via analytical derivative/response theory methods. For the analytic response approach based on CC, please see AUTOCI Response Properties via Analytic Derivatives. For evaluation of spin Hamiltonian parameters using quasi-degenerate perturbation theory, you may be interested in Magnetic Properties Through Quasi Degenerate Perturbation Theory A complete list of possible keywords for the eprnmr module can be found in EPRNMR - keywords for magnetic properties. The cartesian index convention for EPR can be found in Cartesian Index Conventions for EPR and NMR Tensors.
5.24.1. The g-Tensor¶
The g-tensor determines the position of the signal in an EPR spectrum. It can be calculated as a second derivative of the energy and it is implemented as such in ORCA for the SCF methods (HF and DFT), CASSCF, as well as all-electron MP2 (or RI-MP2) and double-hybrid DFT, plus Coupled Cluster (CCSD and CCSD(T)).
5.24.1.1. Theory¶
The g-tensor has four contributions which may be evaluated in ORCA,
These contributions are:
The spin Zeeman contribution, which is isotropic and characterized by a simple constant, the free-electron g-value
(5.104)¶meaning that it doesn’t require any computation. The remaining terms contribute to the deviation from the free-electron value, which is sometimes called the g-shift.
The first-order relativistic mass correction, which is a usually rather small scalar-relativistic correction and can be evaluated as an expectation value over the spin density
(5.105)¶The diamagnetic spin-orbit (also called gauge correction) term, another first-order correction which is often small
(5.106)¶The second-order orbital Zeeman/SOC contribution, usually the main source of deviation from the free-electron g-value in a molecule,
(5.107)¶Its calculation requires the solution of response equations to obtain the perturbed spin density, this is done for the magnetic field perturbation in ORCA. Precise details depend on the level of theory. At the SCF level, it is achieved by solving coupled-perturbed SCF equations (see CP-SCF Options). At the CASSCF level, the CP-CASSCF equations are solved (CASSCF Linear Response).
In these equations,
5.24.1.2. Basic usage¶
As an example, consider the following simple g-tensor job:
! BP86 Def2-SVP TightSCF g-tensor SOMF(1X)
* int 1 2
O 0 0 0 0 0 0
H 1 0 0 1.1056 0 0
H 1 2 0 1.1056 109.62 0
*
The simplest way is to call the g-tensor property in the simple input
line as shown above. It can also be specified in the %eprnmr
block
with gtensor true
. SOMF(1X)
defines the chosen spin-orbit coupling
(SOC) operator.
The output looks like the following. It contains information on the contributions to the g-tensor (relativistic mass correction, diamagnetic spin-orbit term (= gauge-correction), paramagnetic spin-orbit term (= OZ/SOC)), the isotropic g-value and the orientation of the total tensor.
-------------------
ELECTRONIC G-MATRIX
-------------------
The g-matrix:
2.0104321 -0.0031354 -0.0000000
-0.0031354 2.0081968 -0.0000000
-0.0000000 -0.0000000 2.0021275
Breakdown of the contributions
gel 2.0023193 2.0023193 2.0023193
gRMC -0.0003174 -0.0003174 -0.0003174
gDSO(tot) 0.0000808 0.0001539 0.0001515
gPSO(tot) 0.0000449 0.0038301 0.0104898
---------- ---------- ----------
g(tot) 2.0021275 2.0059858 2.0126431 iso= 2.0069188
Delta-g -0.0001917 0.0036665 0.0103238 iso= 0.0045995
Orientation:
X 0.0000000 0.5762906 -0.8172448
Y 0.0000000 0.8172448 0.5762906
Z 1.0000000 -0.0000000 0.0000000
g-tensor calculations at the SCF level are not highly demanding in terms of basis set size. Basis sets that give reliable SCF results (at least valence double-zeta plus polarization) usually also give reliable g-tensor results. For many molecules the Hartree-Fock approximation will give reasonable predictions. In a number of cases, however, it breaks down completely. DFT is more robust and the number of molecules where it fails is much smaller. Among the density functionals, the hybrid functionals seem to be the most accurate.
Note
There are different options for treatment of the spin-orbit coupling operator available in the program. The defaults should be quite reliable. Check out The Spin-Orbit Coupling Operator for more information.
5.24.1.3. Gauge origin treatment¶
The g-tensor is not gauge-independent. Without taking special precautions, the results will have an unphysical dependence on where the gauge origin is placed with respect to the molecule. Unless a fully invariant procedure (such as GIAOs) is used, this undesirable aspect is always present in the calculations.
Starting from ORCA 5.0, the default treatment for g-tensor calculations
gauge is the GIAO approach (GIAO stands for “gauge including atomic orbitals”).
GIAOs are available at the SCF level, RI-MP2 as well as Coupled Cluster up to CCSD(T) level. GIAOs are not currently available with CASSCF linear response
and a common gauge origin must be provided in the %eprnmr
block.
The GIAO one-electron integrals are done analytically by
default whereas the treatment of the GIAO two-electron integrals is
chosen to be same as for the SCF. The available options which can be set
with giao_1el / giao_2el
in the %eprnmr
block can be found in
section EPRNMR - keywords for magnetic properties.
Concerning the computational time, for small systems, e.g. phenyl
radical (41 electrons), the rijk
-approximation is good to use for the
SCF-procedures as well as the GIAO two-electron integrals. Going to
larger systems, e.g. chlorophyll radical (473 electrons), the
rijcosx
-approximation reduces the computational time enormously. While
the new default grid settings in ORCA 5.0 (defgrid2
) should be
sufficient in most cases, certain cases might need the use of
defgrid3
. Note that for the current implementation of CC-NMR/EPR, both
giao_1el
and giao_2el
need to be evaluated fully analytically, which is also
the default when CC-NMR/EPR is requested.
If the choice of the gauge origin is not outrageously poor, a common gauge
origin often gives reasonable results for g-tensors (much more reasonable than
in NMR, where it shouldn’t be used). This is especially true with a large basis set.
The origin may be modified with the keyword ori
inside the %eprnmr
input block.
If you are using a common gauge origin, it is wise to check the sensitivity of the
results with respect to origin location, especially when small
g-shifts on the order of only a few hundred ppm are calculated.
5.24.1.4. Keywords¶
Keyword |
Options |
Description |
---|---|---|
|
|
Calculate the g-tensor |
|
|
Print the 1- and 2-electron contributions to the g-tensor |
|
|
use the GIAO formalism (default) |
|
Common gauge origin at center of electronic charge |
|
|
Common gauge origin at center of nuclear charge |
|
|
Common gauge origin at center of spin density |
|
|
Common gauge origin at center of mass |
|
|
Common gauge origin at atom |
|
|
Common gauge origin at position |
|
|
|
whether to include the 2-el contribution to GIAO term from the SOMF operator (usually small but expensive, disabled by default) |
5.24.2. Hyperfine Coupling¶
Hyperfine couplings characterize the interaction between the electronic spin and the spin of a nucleus in the molecule. In typical EPR spectra, they lead to a splitting of the signal.
5.24.2.1. Theory¶
The hyperfine coupling has four contributions which may be evaluated in ORCA,
These contributions are:
The isotropic Fermi contact term that arises from finite spin density on the nucleus under investigation. It is calculated for nucleus
from:(5.109)¶Here,
is the expectation value of the z-component of the total spin, and are the electron and nuclear g-factors and and are the electron and nuclear magnetons respectively. is the spin density at the nucleus. The proportionality factor is commonly used and has the dimensions MHz bohr in ORCA.The spin dipole term that arises from the through-space dipole-dipole interaction of the magnetic nucleus with the magnetic moment of the electron. It is also calculated as an expectation value over the spin density as:
(5.110)¶where
is the spin-density matrix and is a vector of magnitude that points from the nucleus in question to the electron ( is the set of basis functions).The second-order spin-orbit contribution, which arises as a cross-term between the spin-orbit and nucleus-orbit coupling operators. It requires the solution of coupled-perturbed SCF equations and is consequently more computationally demanding. The contribution can be written as:
(5.111)¶The derivative of the spin density is computed by solving the coupled-perturbed SCF equations with respect to the nucleus-orbit coupling perturbation, which is represented by the operator
(5.112)¶where the sum is over electrons and
is the nucleus in question.The gauge-correction contribution (sometimes also called diamagnetic). This term is often small. However, it is needed in order to get exactly gauge-invariant results. We recently showed that the gauge correction can become crucial in the long-distance limit between the nuclear spin and the electron spin. This is relevant for pseudocontact NMR chemical shifts (PCS).[702]
Note
Second-order HFCs can be quite significant for heavier nuclei and are certainly good to include for transition metal complexes. Available treatments of the spin-orbit coupling operator are described under The Spin-Orbit Coupling Operator.
5.24.2.2. Basic usage¶
Hyperfine and quadrupole couplings can be requested in the %eprnmr
input block.
Since there may be several nuclei that you are interested in, the input is
relatively sophisticated.
An example how to calculate the hyperfine and field gradient tensors for the CN radical is given below:
! PBE0 def2-MSVP TightSCF
* int 0 2
C 0 0 0 0 0 0
N 1 0 0 1.170 0 0
*
%eprnmr Nuclei = all C { aiso, adip }
Nuclei = 2 { aiso, adip, fgrad }
end
In this example, the hyperfine tensor is calculated for all carbon atoms and atom 2, which is nitrogen in this case. (Additionally, the electric field gradient is calculated for nitrogen.)
Note
Counting of atom numbers starts from 1.
Beware - all nuclei mentioned in one line will be assigned the same isotopic mass! If several different nuclei are desired (such as C and H), there has to be a new line for each of them.
You have to specify the
Nuclei
statement after the definition of the atomic coordinates or the program will not figure out what is meant by “all
”.
The output looks like the following. It contains detailed information about the individual contributions to the hyperfine couplings, its orientation, its eigenvalues, the isotropic part and (if requested) also the quadrupole coupling tensor.
-----------------------------------------
ELECTRIC AND MAGNETIC HYPERFINE STRUCTURE
-----------------------------------------
-----------------------------------------------------------
Nucleus 0C : A:ISTP= 13 I= 0.5 P=134.1903 MHz/au**3
Q:ISTP= 13 I= 0.5 Q= 0.0000 barn
-----------------------------------------------------------
Raw HFC matrix (all values in MHz):
-----------------------------------
695.8952 0.0000 -0.0000
0.0000 543.0617 -0.0000
-0.0000 -0.0000 543.0617
A(FC) 594.0062 594.0062 594.0062
A(SD) -50.9445 -50.9445 101.8890
---------- ---------- ----------
A(Tot) 543.0617 543.0617 695.8952 A(iso)= 594.0062
Orientation:
X 0.0000000 0.0000000 -1.0000000
Y -0.8111216 -0.5848776 -0.0000000
Z -0.5848776 0.8111216 0.0000000
Notes: (1) The A matrix conforms to the "SAI" spin Hamiltonian convention.
(2) Tensor is right-handed.
-----------------------------------------------------------
Nucleus 1N : A:ISTP= 14 I= 1.0 P= 38.5677 MHz/au**3
Q:ISTP= 14 I= 1.0 Q= 0.0204 barn
-----------------------------------------------------------
Raw HFC matrix (all values in MHz):
-----------------------------------
13.2095 -0.0000 0.0000
-0.0000 -45.6036 -0.0000
0.0000 -0.0000 -45.6036
A(FC) -25.9993 -25.9993 -25.9993
A(SD) 39.2088 -19.6044 -19.6044
---------- ---------- ----------
A(Tot) 13.2095 -45.6036 -45.6036 A(iso)= -25.9993
Orientation:
X 1.0000000 0.0000000 -0.0000000
Y -0.0000000 0.9996462 0.0265986
Z 0.0000000 -0.0265986 0.9996462
Notes: (1) The A matrix conforms to the "SAI" spin Hamiltonian convention.
(2) Tensor is right-handed.
Raw EFG matrix (all values in a.u.**-3):
-----------------------------------
-0.1832 -0.0000 0.0000
-0.0000 0.0916 0.0000
0.0000 0.0000 0.0916
V(El) 0.6468 0.6468 -1.2935
V(Nuc) -0.5551 -0.5551 1.1103
---------- ---------- ----------
V(Tot) 0.0916 0.0916 -0.1832
Orientation:
X -0.0000003 0.0000002 1.0000000
Y 0.9878165 0.1556229 0.0000003
Z -0.1556229 0.9878165 -0.0000002
Note: Tensor is right-handed
Quadrupole tensor eigenvalues (in MHz;Q= 0.0204 I= 1.0)
e**2qQ = -0.880 MHz
e**2qQ/(4I*(2I-1))= -0.220 MHz
eta = 0.000
NOTE: the diagonal representation of the SH term I*Q*I = e**2qQ/(4I(2I-1))*[-(1-eta),-(1+eta),2]
If also EPR g-tensor or D-tensor calculations (see next section) are carried out in the same job, ORCA automatically prints the orientation between the hyperfine/quadrupole couplings and the molecular g- or D-tensor. For more information on this see section orca_euler.
Note
For heavy nuclei you may want to consider the possibility of relativistic effects. Scalar relativistic effects can be handled with several quasi-relativistic Hamiltonians in ORCA. An overview of the possibilities and some recommendations can be found in Relativistic Calculations. Note that relativistic calculations may have special requirements on basis sets. In relativistic property calculations, you should be aware of the importance of picture change corrections (see Picture-Change Effects). In quasi-relativistic calculations with DFT, one should also be very cautious about accuracy of the numerical integration, especially for heavier (transition metal) nuclei.
Note
For the calculation of HFCCs using DLPNO-CCSD, it is recommended to use
the tailored truncation settings !DLPNO-HFC1
or !DLPNO-HFC2
in the
simple keyword line which correspond to the “Default1” and “Default2”
setting in Ref. [703].
If you wish to extract the A tensor for an oligonuclear transition metal complex, the
A(iso)
value in the output can be processed according to the method described
in ref. [704].
5.24.2.3. A note on basis sets¶
For hyperfine (and quadrupole) couplings, standard basis sets designed for energies and geometry optimizations and are often not satisfactory (especially for atoms heavier than Ne). You should look into a basis set capable of describing the electronic structure near the nucleus. One option is to use a basis set tailored towards “core-property” calculations. The following are good options:
EPR-II
basis of Barone and co-workers: It is only available for a few light atoms (H, B, C, N, O, F) and is essentially of double-zeta plus polarization quality with added flexibility in the core region, which should give reasonable results.IGLO-II
andIGLO-III
bases of Kutzelnigg and co-workers: They are fairly accurate but also only available for some first and second row elements.CP basis: They are accurate for first row transition metals as well.
uncontracted Partridge basis: They are general purpose HF-limit basis sets and will probably be too expensive for routine use, but are very useful for calibration purposes.
the
pcH
basis sets by Jensen tailored towards hyperfine coupling calculations, also with limited availability (see Jensen Basis Sets).
If ORCA does not yet have a dedicated basis set for your element, you will likely
have to tailor the basis set to your needs. You can
do this by manually adding s-type primitives with large exponents, or by
decontracting core parts of the basis set (see Basis Sets
for description of the decontraction feature).
You can start by examining the basis set you already have - if you
add the statement Print[p_basis] 2
in the %output
block
(or PrintBasis
in the simple input line) the program
will print the basis set in input format (for the basis block).
You can then add or remove primitives, uncontract core functions, etc. For
example, here is a printout of the carbon basis DZP in input format:
# Basis set for element : C
NewGTO 6
s 5
1 3623.8613000000 0.0022633312
2 544.0462100000 0.0173452633
3 123.7433800000 0.0860412011
4 34.7632090000 0.3022227208
5 10.9333330000 0.6898436475
s 1
1 3.5744765000 1.0000000000
s 1
1 0.5748324500 1.0000000000
s 1
1 0.1730364000 1.0000000000
p 3
1 9.4432819000 0.0570590790
2 2.0017986000 0.3134587330
3 0.5462971800 0.7599881644
p 1
1 0.1520268400 1.0000000000
d 1
1 0.8000000000 1.0000000000
end;
Reading this information is relatively straightforward.
For example: the “s 5
” stands for the angular momentum and the number
of primitives in the first basis function. The following five lines
then contain the number of the primitive, the exponent and the contraction
coefficient (unnormalized) for each primitive in the function.
Remember that if you add very steep functions, you must increase the
size of the integration grid if your calculation uses DFT! Otherwise,
your results will be inaccurate. To some extent, the program now
automatically adapts the grids when it detects steep functions.
If you further need to increase integration accuracy, you could globally
increase the radial grid size by increasing IntAcc
in the Method
block, or increase integration accuracy for individual atoms
only, with the latter option being less expensive. More detail can be found
in section Other Details and Options.
In the present example the changes caused by larger basis sets in the core
region and more accurate integration are relatively modest – on the order
of 3%. This is, however, still significant if you are a little puristic.
5.24.2.4. Hyperfine Coupling Keywords¶
Warning
All nuclei specified on one line will be assigned with the same isotopic mass (and other parameters)!
Keyword |
Selection |
What to do |
Description |
---|---|---|---|
|
|
Selects all nuclei of element |
|
|
Selects atoms |
||
|
Calculates the isotropic Fermi-contact contribution |
||
|
Calculates the spin-dipole contribution |
||
|
Calculates the spin-orbit contribution |
||
|
Selects isotope |
||
|
Sets HFC proportionality factor |
||
|
Sets nuclear spin to |
||
|
example requesting multiple actions for selection |
Keyword |
Options |
Description |
---|---|---|
|
|
use effective nuclear charges for the gauge correction |
|
|
calculate DSO integrals numerically (faster) |
|
|
< 0 means to use the DFT grid setting |
|
|
< 0 means to use the DFT grid setting |
|
|
< 0 means to use the DFT grid setting |
|
|
< 0 means to use the DFT grid setting |
|
|
< 0 means to use the DFT grid setting |
5.24.3. Zero-Field Splitting¶
The zero-field splitting (ZFS) is typically the leading term in the
Spin-Hamiltonian (SH) for transition metal complexes with a total ground
state spin
5.24.3.1. Theory¶
There are two important contributions to zero-field splitting [705]:
A first order term arising from the direct spin-spin interaction
(5.113)¶where
, x,y,z, is the fine structure constant ( 1/137 in atomic units), is the electronic distance vector with magnitude and is the spin-vector operator for the ’th electron. is the exact ground state eigenfunction of the Born-Oppenheimer Hamiltonian with total spin and projection quantum number . Since the spin-spin interaction is of first order, its evaluation presents no particular difficulties.A second-order spin-orbit contribution, which is substantially more complicated. Under the assumption that the spin-orbit coupling (SOC) operator can to a good approximation be represented by an effective one-electron operator (
, ref [656] has derived the following sum-over-states (SOS) equations for the SOC contribution to the ZFS tensor:(5.114)¶(5.115)¶(5.116)¶Here the one-electron spin-operator for electron
has been written in terms of spherical vector operator components with and is the excitation energy to the excited state multiplet (all components are degenerate at the level of the BO Hamiltonian).
Note
The second-order spin-orbit term may be evaluated using the above sum-over-states formulation in the context of wavefunction theory methods. The following section focuses on a formulation based on SCF analytical derivative/response theory. You may also be interested in QDPT, which is more reliable particularly in systems where zero-field splitting is large.
Note
Available treatments of the spin-orbit coupling operator are described under The Spin-Orbit Coupling Operator.
5.24.3.2. Basic usage¶
For example, consider the following job on a hypothetical Mn(III)-complex.
! BP86 def2-SVP SOMF(1X)
%eprnmr DTensor ssandso
DSOC cp # qro, pk, cvw
DSS uno # direct
end
* int 1 5
Mn 0 0 0 0 0 0
O 1 0 0 2.05 0 0
O 1 2 0 2.05 90 0
O 1 2 3 2.05 90 180
O 1 2 3 2.05 180 0
F 1 2 3 1.90 90 90
F 1 2 3 1.90 90 270
H 2 1 6 1.00 127 0
H 2 1 6 1.00 127 180
H 3 1 6 1.00 127 0
H 3 1 6 1.00 127 180
H 4 1 6 1.00 127 0
H 4 1 6 1.00 127 180
H 5 1 6 1.00 127 0
H 5 1 6 1.00 127 180
*
The output documents the individual contributions to the D-tensor, which also contains (unlike the g-tensor) contributions from spin-flip terms.
Note
There are four different variants of the SOC-contribution, which alone should demonstrate that this is a difficult property. The options are:
The QRO method is fully documented[360] and is based on a theory developed earlier.[656] The QRO method is reasonable but somewhat simplistic and is superseded by the CP method described below.
The Pederson-Khanna model was brought forward in 1999 from qualitative reasoning.[706] It also contains incorrect prefactors for the spin-flip terms. We have nevertheless implemented the method for comparison. In the original form it is only valid for local functionals. In ORCA it is extended to hybrid functionals and HF.
The default coupled-perturbed method is a generalization of the DFT method for ZFSs; it uses revised prefactors for the spin-flip terms and solves a set of coupled-perturbed equations for the SOC perturbation. Therefore it is valid for hybrid functionals. It has been described in detail.[707]
The present implementation in ORCA is valid for HF, DFT and hybrid DFT.
The DSS part is an expectation value that involves the spin density of the system. In detailed calibration work[708] it was found that the spin-unrestricted DFT methods behave somewhat erratically and that much more accurate values were obtained from open-shell spin-restricted DFT. Therefore the “UNO” option allows the calculation of the SS term with a restricted spin density obtained from the singly occupied unrestricted natural orbitals.
The DSS part contains an erratic self-interaction term for UKS/UHF
wavefunction and canonical orbitals. Thus, UNO
is recommended for
these types of calculations.[709] If the option
DIRECT
is used nevertheless, ORCA will print a warning in the
respective part of the output.
Note
In case that D-tensor is calculated using the correlated wave
function methods such as (DLPNO-/LPNO-)CCSD, one should not use
DSS=UNO
option.
5.24.3.3. Detailed theory of the coupled-perturbed formulation¶
In 2007, we have developed a procedure that makes the ZFS calculation
compatible with the language of analytic derivatives.[707]
Perhaps the most transparent route is to start from the exact solutions
of the Born-Oppenheimer Hamiltonian. To this end, we look at the second
derivative of the ground state energy
(
Where
With the components of the spin density:
The second derivative with respect to a second component for
The derivative of the spin density may be written as:
Expanding the perturbed wavefunction in terms of the unperturbed states gives to first order:
Where
The equality holds for exact states. For approximate electronic structure treatments, the analytic derivative approach is more attractive since an infinite sum over states can never be performed in practice and the calculation of analytic derivative is computationally less demanding than the calculation of excited many electron states.
Using eq. (5.123), the components of the SOC-contribution to the D-tensor are reformulated as
These are general equations that can be applied together with any non-relativistic or scalar relativistic electronic structure method that can be cast in second quantized form. Below, the formalism is applied to the case of a self-consistent field (HF, DFT) reference state.
For DFT or HF ground states, the equations are further developed as follows:
The SCF energy is:
Here
with MO coefficients
The mixing parameter
With
For the second derivative, the perturbed orbitals are required. However,
in the presence of a spin-dependent perturbation they can no longer be
taken as pure spin-up or spin-down orbitals. With respect to the
Since the matrix elements of the spin-vector operator components are purely real and the spatial part of the SOC operator has purely imaginary matrix elements, it follows that the first order coefficients are purely imaginary. The second derivative of the total SCF energy becomes:
Examination of the three cases
Where a special form of the perturbed densities has been chosen. They are given in the atomic orbital basis as:
The special form of the coupled perturbed equations are implemented in
ORCArun as follows: The perturbation is of the general form
Note that these coupled-perturbed (CP) equations contain no contribution from the Coulomb potential or any other local potential such as the exchange-correlation potential in DFT. Hence, in the absence of HF exchange, the equations are trivially solved:
It is interesting that the “reverse spin flip coefficients”
Equations (5.142)-(5.147) are referred to as CP-SOC (coupled-perturbed spin-orbit coupling) equations. They can be solved by standard techniques and represent the desired analogue of the CP-SCF magnetic response equations solved for the determination of the g-tensor and discussed in detail earlier [710]. It is readily confirmed that in the absence of HF exchange, eqs. (5.148)-(5.153) inserted into eqs. (5.136)-(5.141) lead back to a modified Pederson-Khanna type treatment of the SOC contributions to the D-tensor [206]. In the framework of the formalism developed above, the Pederson-Khanna formula can be re-written in the form:
This equation was derived from second-order non-self-consistent
perturbation theory without recourse to spin-coupling. For the special
case of no Hartree-Fock exchange, the main difference to the treatment
presented here is that the correct prefactors from eqs.
(5.125)-(5.127) occur in front of the spin-flip contributions
rather than
For completeness, the evaluation of the spin-spin term in the SCF case proceeds conveniently through:
as derived by McWeeny and Mizuno and discussed in some detail by Sinnecker and Neese.[708] In this reference it was found that DFT methods tend to overestimate the spin-spin contribution if the calculations are based on a spin-unrestricted SCF treatment. A much better correlation with experiment was found for open-shell spin restricted calculations. The origin of this effect proved to be difficult to understand but it was shown in ref [207] that in the case of small spin-contamination, the results of ROKS calculations and of those that are obtained on the basis of the spin-unrestricted natural orbital (UNO) determinant are virtually indistinguishable. It is therefore optionally possible in the ORCA program to calculate the spin-spin term on the basis of the UNO determinant.
5.24.3.4. ZFS Keywords¶
Keyword |
Options |
Description |
---|---|---|
|
|
Calculate spin-spin part |
|
Calculate spin-orbit part |
|
|
Calculate both parts |
|
|
|
quasi-restricted method for SOC, must be combined with |
|
Pederson-Khanna method for SOC |
|
|
coupled-perturbed method for SOC (default) |
|
|
van Wüllen method for SOC |
|
|
|
directly use the canonical orbitals for the spin density |
|
use spin density from UNOs |
|
|
|
Whether to calculate Euler angles via |