7.43. Magnetic properties through Quasi Degenerate Perturbation Theory¶
7.43.1. Quasi Degenerate Perturbation Theory (QDPT) in a nutshell¶
Quasi Degenerate Perturbation Theory offers a versatile and accurate approach to to a number of magnetic properties for basically every wavefunction based excited states method.
In a nutshell at the non relativistic limit for every excited state single or multireference wavefunction based method, bearing a CASSCF, MRCI, or a ROCIS type of zeroth order wavefunction one can set up an excitation problem that is a combination of the zeroth order wavefunction and excited spin-adapted configuration state functions (CSFs)\(\left| \Phi_{\mu}^{SS} \right \rangle\).
That takes the form:
Here the upper indices \(SS\) stand for a wave function of the spin quantum number \(S\) and spin projection \(M_{S} = S\). Since the BO Hamiltonian does not contain any complex-valued operator, the solutions \(\left| \Psi_{I}^{SS}\right\rangle\) may be chosen to be real-valued.
By obtaining a solution to the above eigenvalue problem provides the coefficients with which the CSFs enter into the chosen wavefunction as well as the eigenstates of the spin-free operator. These eigenstates maube used to expand towards the respective relativistic eigenstates by setting up the relevant quasi-degenerate eigenvalue problem. In fact the spin-orbit coupling (SOC), the spin-spin coupling (SSC) effects along with the Zeeman interaction can be included by means of the quasi-degenerate perturbation theory (QDPT). In this approach the SOC, the SSC, and the Zeeman operators are calculated in the basis of pre-selected solutions of the BO Hamiltonian\(\left\{{ \Psi_{I}^{SM} } \right\}\).
Diagonalization of this matrix yields the energy levels and eigenvectors of the coupled states. These eigenvectors in fact represent linear combinations of the solutions of \(\hat{{H} }_{\text{BO} }\) with complex coefficients.
The effective one-electron SOC operator in second quantized form can be written as [611]:
Here \(\hat{{a} }_{p}^{\uparrow }\) and \(\hat{{b} }_{p}^{\uparrow }\) stand for creation of \(\alpha\) and \(\beta\) electrons respectively; \(\hat{{a} }_{p}\) and \(\hat{{b} }_{p}\) represent the corresponding annihilation operators. The matrix elements \(z_{pq}^{-} = z_{pq}^{x} - iz_{pq}^{y}\), \(z_{pq}^{+} = z_{pq}^{x} + iz_{pq}^{y}\), and \(z_{pq}^{0} = z_{pq}^{z}\) (upper \(x\), \(y\), \(z\) indices denote the Cartesian components) are constructed from the matrix elements described in section Zero-Field-Splitting.
In this concept the SOC Hamiltonian reads:
where \(m\) represents the standard vector operator components. \(\left( \begin{matrix}S^\prime&1\\M^\prime&m\\\end{matrix} \left|{\begin{matrix} S\\ M\\ \end{matrix} } \right. \right)\) is a Clebsch–Gordon coefficient that has a single numer- ical value that is tabulated. It satisfies certain selection rules and contains all of the M-dependence of the SOC matrix elements. The quantity \(Y_{II^\prime}^{SS^\prime}(m)\) is a reduced matrix element. It only depends on the standard components of the two states involved. There are only three cases of non-zero \(Y_{II^\prime}^{SS^\prime}(m)\) which arise from state pairs that either have the same total spin or differ by one unit.[622]
The SSC Hamiltonian reads:
For matrix elements between states of the same multiplicity it can be simplified to
Here
represents the two-electron quintet density. The operators \(E_{pq} =\hat{{a} }_{p}^{\uparrow } \hat{{a} }_{q} +\hat{{b} }_{p}^{\uparrow } \hat{{b} }_{q}\) and \(S_{pq}^{z} =\hat{{a} }_{p}^{\uparrow } \hat{{a} }_{q} -\hat{{b} }_{p}^{\uparrow } \hat{{b} }_{q}\) symbolize here the one-electron density operator and the spin density operator accordingly. The spatial part
denotes the two-electron field gradient integrals. These two-electron integrals can be evaluated using the RI approximation.
Finally, the Zeeman Hamiltonian is included in the form of:
with \(\mathrm{\mathbf{\hat{L} }}\) representing the total orbital momentum operator, and \(\mathrm{\mathbf{\hat{S} }}\) being the total spin operator.
In this concept solution of a selected relativistic Hamiltonian provide access to a numerous magnetic properties namely EPR properties EPR and NMR properties as well as Magnetization and Susceptibility properties Magnetization and Magnetic Susceptibility In addition monitoring the impact of an external Magnetic Field to the relativistic eigenstates and eigenvectors Addition of Magnetic Fields becomes straightforward.
Collectively within the QDPT framework the following magnetic properties become available
G-Tensor/Matrix
Zero Field Splitting
Hyperfine A-Tensor/Matrix
Electric Field Gradient
Magnetization
Susceptibility
Inclusion of Magnetic Fields
7.43.2. Magnetic properties through the Effective Hamiltonian¶
Since both the energies and the wavefunction of the low-lying spin-orbit states are available, the effective Hamiltonian theory can be used to extract EPR parameters such as the full G, Zero Field Splitting (ZFS) and hyperfine A tensors.
Provided that the ground state is non-degenerate. By applying this Hamiltonian on the basis of the model space, i.e. the \(|S, M_S\rangle\) components of the ground state, the interaction matrix is constructed.
The construction of effective Hamiltonian relies on the information contained in both the energies and the wavefunctions of the low-lying spin-orbit states. Following des Cloizeaux formalism, the effective Hamiltonian reproduces the energy levels of the “exact” Hamiltonian \(E_k\) and the wavefunctions of the low-lying states projected onto the model space \(\tilde{\Psi}\):
These projected vectors are then symmetrically orthonormalized resulting in an Hermitian effective Hamiltonian, which can be written as:
The effective interaction matrix obtained by expanding this Hamiltonian into the basis of determinants belonging to the model space, is the compared to the matrix resulted from expanding the model Hamiltonian. Based on a singular value decomposition procedure, all 9 elements of the G, A and/or ZFS tensors may be extracted.
7.43.3. Organization of QDPT Magnetic Properties Computation¶
Starting from ORCA 6.0 the calculation of the magnetic properties through the
Quasi Degenerate Perturbation Theory (QDPT) in all available correlation type modules is unified
and simplified. Following the general architecture design of ORCA 6.0 the computation
of all the involved magnetic properties are centrally performed by a driver data structure called the QDPT Driver.
The Driver takes into account all the specific variables that are populated by the involved module
and proceeds accordingly to calculated and represent the requested property in a uniform fashion.
This presently involves the casscf, mrci, rocis and lft modules
In this way
Results analysis process from the user’s perspective is simplified
Cross module correlation and comparisons are also easily accessible
The general keywords that activate the generation of QDPT properties are:
%method (casscf, mrci, lft, ...)
relativistic block (soc, rel, ...)
DoSOC true # include the SOC contribution
DoSSC true # include the SSC contribution
end
end
In a first step SOC contributions will be computed for any level of theory that is available
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QDPT WITH CASSCF/NEVPT2/MRCI... DIAGONAL ENERGIES
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*************************************
COMPUTING QDPT HAMILTONIAN
*************************************
*************************************
Doing QDPT with ONLY SOC!
*************************************
Upon request the non-zero SOC Matrix Elemnts will be printed
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NONZERO SOC MATRIX ELEMENTS (cm**-1)
------------------------------------
Bra Ket
<Block Root S Ms | HSOC | Block Root S Ms> = Real-part Imaginary part
--------------------------------------------------------------------------------------
0 2 1.0 1.0 0 1 1.0 1.0 0.000 -71.172
0 3 1.0 1.0 0 2 1.0 1.0 0.000 1.542
0 4 1.0 1.0 0 0 1.0 1.0 0.000 50.048
0 5 1.0 1.0 0 0 1.0 1.0 0.000 -48.827
0 5 1.0 1.0 0 4 1.0 1.0 0.000 -40.119
0 6 1.0 1.0 0 1 1.0 1.0 0.000 -0.197
0 6 1.0 1.0 0 3 1.0 1.0 0.000 8.724
0 7 1.0 1.0 0 1 1.0 1.0 0.000 -2.695
followed by printing of the SOC Hamiltonian
Note: In the following the full <I|HBO+SOC|J> are printed in the CI Basis.
I,J are compound indices for |Block/Mult, Ms, Root>, where the states
are ordered first by MultBlock, then Ms and finally Root.
-----------------
SOC MATRIX (A.U.)
-----------------
leading to the printout of the relativistically corrected eigenvalues and eigenvectors
Lowest eigenvalue of the SOC matrix: -149.86223277 Eh
Energy stabilization: -2.54512 cm-1
Eigenvalues: cm-1 eV Boltzmann populations at T = 300.000 K
0: 0.00 0.0000 3.36e-01
1: 2.37 0.0003 3.32e-01
2: 2.37 0.0003 3.32e-01
3: 7757.65 0.9618 2.33e-17
4: 7757.66 0.9618 2.33e-17
5: 11913.81 1.4771 5.15e-26
...
The threshold for printing is 0.0100
Eigenvectors:
Weight Real Image : Block Root Spin Ms
STATE 0: 0.0000
0.388265 0.410320 -0.468937 : 0 0 1 1
0.223270 -0.000000 0.472514 : 0 0 1 0
0.388265 0.410320 0.468937 : 0 0 1 -1
STATE 1: 2.3703
0.310686 0.534586 0.157809 : 0 0 1 1
0.378606 0.000017 -0.615309 : 0 0 1 0
0.310706 0.534623 -0.157747 : 0 0 1 -1
STATE 2: 2.3703
0.300970 -0.214003 -0.505146 : 0 0 1 1
0.398078 -0.000007 -0.630934 : 0 0 1 0
0.300949 -0.214019 0.505119 : 0 0 1 -1
...
Then in following all the relevant QDPT properties will be printed:
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COMPUTING QDPT PROPERTIES
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1) for G-Tensor/Matrix
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ELECTRONIC G-MATRIX FROM EFFECTIVE HAMILTONIAN
----------------------------------------------
2) for ZFS, on the basis of the 2nd Order and Effective Hamiltonian approximations
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ZERO-FIELD SPLITTING
2ND ORDER SOC CONTRIBUTION
--------------------------------------------
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ZERO-FIELD SPLITTING
EFFECTIVE HAMILTONIAN SOC CONTRIBUTION
--------------------------------------------------------
3) for A-Tensor/Matrix
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QDPT HFC A-MATRICES
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4) for Electric Field Gradient Tensor
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EFG TENSOR
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5) For Mahnetization/Susceptibility
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SOC CORRECTED MAGNETIZATION AND/OR SUSCEPTIBILITY
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6) For External Magnetic Fields Contributions
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SOC TRANSITION MAGNETIC DIPOLE CONTRIBUTIONS IN EXTERNAL MAGNETIC FIELD
Magnetic field Bx = 1.00 Gauss By = 0.00 Gauss Bz = 0.00 Gauss
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States Energy Energy Osh.Str M2 MX MY MZ
(cm-1) (eV) (au) (au**2) (au) (au) (au)
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0 0 0.00 0.0000 0.00000000 0.00000145 0.00090159 0.00061858 0.00050413
0 1 20.20 0.0025 0.00000000 0.00000126 0.00042808 0.00067630 0.00078545
1 1 0.00 0.0000 0.00000000 0.00000000 0.00003430 0.00002347 0.00001909
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Following this as Discussed in One Photon Spectroscopy Section {ref}`sec:ops.detailed` all relativistically corrected optical spectra will also printed under the same correction scheme
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SOC CORRECTED ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS
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Transition Energy Energy Wavelength fosc(D2) D2 |DX| |DY| |DZ|
(eV) (cm-1) (nm) (*population) (au**2) (au) (au) (au)
--------------------------------------------------------------------------------------------------------
...
If requested in a second step Spin-Spin Coupling contributions will be generated and wiil be added to the SOC Hamiltonian to generate SOC+SSC contributions
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Calculating Spin-Spin Coupling Integrals
-------------------------------------------
The the program will undergo the exact same analysis as above printing the SOC+SSC analysis
***********************************************************
* DOING EVERYTHING A SECOND TIME: THIS TIME INCLUDING SSC *
***********************************************************
*************************************
COMPUTING QDPT HAMILTONIAN
*************************************
*************************************
Doing QDPT with SOC AND SSC!
*************************************
------------------------------------
NONZERO SOC and SSC MATRIX ELEMENTS (cm**-1)
------------------------------------
Bra Ket
<Block Root S Ms | HSOC + HSSC | Block Root S Ms> = Real-part Imaginary part
--------------------------------------------------------------------------------------
0 2 1.0 1.0 0 1 1.0 1.0 -0.000 -71.172
0 3 1.0 1.0 0 1 1.0 1.0 -0.001 0.000
0 3 1.0 1.0 0 2 1.0 1.0 0.020 1.542
0 4 1.0 1.0 0 0 1.0 1.0 -0.222 50.048
0 5 1.0 1.0 0 0 1.0 1.0 -0.228 -48.827
0 5 1.0 1.0 0 4 1.0 1.0 0.332 -40.119
0 6 1.0 1.0 0 1 1.0 1.0 -0.030 -0.197
for both the Magnetic properties e.g. the ZFS
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ZERO-FIELD SPLITTING
EFFECTIVE HAMILTONIAN SOC and SSC CONTRIBUTION
--------------------------------------------------------
as well as the Optical properties
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SOC+SSC CORRECTED ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS
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Transition Energy Energy Wavelength fosc(D2) D2 |DX| |DY| |DZ|
(eV) (cm-1) (nm) (*population) (au**2) (au) (au) (au)
--------------------------------------------------------------------------------------------------------