(sec:mrci.detailed)= # The Multireference Correlation Module (sec:mrci.general.detailed)= ## General Description A number of **uncontracted** multireference approaches are implemented in ORCA and reside in the `orca_mrci` module. All of these approaches start with a reference wavefunction that consists of multiple configurations (orbital occupation patterns). The reference wavefunction defined in the `ref` subblock can be a complete active space (CAS), restricted active space (RAS) or an arbitrary list of configurations. The total wavefunction is constructed by considering single and double excitations out of the reference configurations. These excited configurations are then used to generate configuration state functions (CSF) that have the proper spin and spatial symmetry. The number of wavefunction parameters rapidly grows with the number of reference functions. The orca_mrci module features a set of truncation criteria (`TSel, TPre, TNat`) that help to reduce the number of wavefunction parameters. Furthermore, by default, the program only considers reference configurations that already have the target spin and spatial symmetry. There are situations, where this is undesired and the restrictions can be lifted with the keyword `rejectinvalidrefs false`. For more information on the theory, the program module as well as its usage we recommend the review article by Neese et al.{cite}`neese_advanced_2007`. A tutorial type introduction to the subject is presented in chapter {ref}`sec:mrci.detailed` of the manual and more examples in the CASSCF tutorial. The detailed documentation of all features of the MR-CI and MR-PT module is somewhat premature and at this point only a summary of keywords is given below. A thorough description of all technical and theoretical subtleties must wait for a later version of the manual. The overall scaling of uncontracted approaches is steep. Hence, the methodology is restricted to small reference spaces and small molecules in general. **Note that all integrals must be kept in memory!** Internally contracted multireference approaches such as NEVPT2 do not share these bottlenecks. Aside from NEVPT2, ORCA features a fully internally contracted MRCI (FIC-MRCI) that resides in the `orca_autoci` module. For more details on the FIC-MRCI we refer to section {ref}`sec:autoci.detailed`. ``` %mrci # ----------------------------------------------------------- # Orbital selection # NOTE: The orbitals are used as supplied. Thus, the ORDER of # orbitals is critical. Say you have # nact electrons in the active space # nint electrons in the internal space # nfrozen electrons # * The first nfrozen/2 orbitals will not be included in the CI # * The next nint/2 orbitals will be doubly occupied in all # references # * the nact electrons are distributed over the,say, mact # orbitals according to the active space definitions. # The remaining orbitals are external. # IT IS YOUR RESPONSIBILITY THAT THE ORBITAL ORDERING MAKES # SENSE! # A sensible two-step procedure is: # * generate some orbitals and LOOK AT THEM. Decide which ones # to include in the CI. # * re-read these orbitals with ! MORead NoIter. Perhaps use # the "rotate" feature to reorder the MOs # Then jump right into the CI which is defined in this se- # cond job # # NOTE: the MRCI module respects the %method FrozenCore settings # ----------------------------------------------------------- Loc 0,0,0 # Localize orbitals in the internal (first flag), active # (second flag) and external space (third flag). UseIVOs false # Use improved virtual orbitals in the CI # --------------------------------- # Method selection # --------------------------------- CIType MRCI # Multireference CI (default) MRDDCI1 # Difference dedicated CI 1-degree of freedom MRDDCI2 # Difference dedicated CI 2-degrees of freedom MRDDCI3 # Difference dedicated CI 3-degrees of freedom MRACPF # Average coupled-pair functional MRACPF2 # Modified version of ACPF MRACPF2a # A slightly modified version of ACPF-2a MRAQCC # Average quadratic coupled-cluster MRCEPA_R # Multireference CEPA due to Ruttink MRCEPA_0 # CEPA-0 approximation SORCI # Spectroscopy oriented CI SORCP # Spectroscopy oriented couplet pair approx. MRMP2 # Multireference Moeller-Plesset at second order MRMP3 # Multireference Moeller-Plesset at third order MRMP4 # Multireference Moeller-Plesset at fourth order # but keeping only singles and doubles relative to # the reference configurations. # --------------------------------- # Selection thresholds # --------------------------------- Tsel 1e-6 # Selection threshold for inclusion in the CI based # 2nd order MP perturbation theory <0|H|I>/DE(MP) Tpre 1e-4 # Selection of configurations in the reference space # after the initial diagonalization of the reference # space only configurations with a weight large>Tpre # to any root are included AllSingles false # include ALL SINGLES in the CI. Default is now TRUE!!! # perturbative estimate of the effect of the rejected configurations EunselOpt 0 # no correction 1 # based on the overlap with the 0th order wavefunction 2 # calculation with the relaxed reference space # coefficients. This is the most accurate and only # slightly more expensive # For CIType=MRCI,MRDDCI and SORCI the approximate correction for # higher excitations DavidsonOpt Davidson1 # default Davidson2 # modified version Siegbahn # Siegbahn's approximation Pople # Pople's approximation # For MRACPF,MRACPF2,MRAQCC and SORCP NelCorr 0 # Number of electrons used for computing the average coupled- # pair correction. # =0 : set equal to ALL electrons in the CI # =-1: set equal to all ACTIVE SPACE electrons # =-2: set equal to ACTIVE SPACE electrons IF inactive doubles # are excluded (as in MRDDCI) # >0 : set equal to user defined input value LinearResponse false # Use ground state correlation energy to compute the shift for # higher roots (not recommended) # --------------------------------- # Natural Orbital Iterations # --------------------------------- NatOrbIters 0 # default # number of average natural orbital iterations Tnat 1e-4 # cutoff of natural orbitals. NOs with an occupation number less # then Tnat will not be included in the next iteration # Also, orbitals with occupation number closer than Tnat to 2.0 # will be frozen in the next iteration Tnat2 -1 # if chosen >0 then Tnat2 is the threshold for freezing the # almost doubly occupied orbitals. Otherwise it is set equal # to Tnat # ---------------------------------- # Additional flags and algorithmic # details # ---------------------------------- PrintLevel 2 # default. Values between 1 and 4 are possible DoDDCIMP2 false # for DDCI calculations: if set to true the program computes # a MP2 like correction for the effect of inactive double # excitations which are not explicitly included in the CI. This # is necessary if you compare molecules at different geometries # or compute potential energy surfaces. # ---------------------------------- # The SORCP model # ---------------------------------- CIType_in # First step CIType CIType_fi # Second step CIType Exc_in # First step excitation scheme Exc_fi # Second step excitation scheme Tsel_in # First step Tsel Tsel_fi # Second step Tsel Tpre_in # First step Tsel Tpre_fi # Second step Tpre # Thus, the SORCI model corresponds to CIType=SORCP with # CIType_in MRCI CIType_fi MRCI # Exc_in DDCI2 Cexc_fi DDCI3 # Tsel_in 1e-5 Tsel_fi 1e-5 # Tpre_in 1e-2 Tpre_fi 1e-2 # ---------------------------------- # Multirerence perturbation theory # ---------------------------------- MRPT_b 0.02 # Intruder state avoidance PT after Hirao (default 0.0) # with this flag individual intruders are shifted away to # to some extent from the reference space MRPT_shift 0.3 # Level shift introduced by Roos which shifts the entire # excited manifold away in order to avoid intruder states. # A correction is applied afterwards but results do depend # on this (arbitrary) value to some extent. H0Opt projected # use an off-diagonal definition of H0 Diagonal # use a diagonal definition of H0 (much faster but maybe # a little less reliable Partitioning MP # Moeller plesset partitioning EN # Epstein-Nesbet partitioning (not recommended) Fopt Standard # Standard definition of MR Fock operators G3 # uses Anderson's g3 correction also used in CASPT2 #--------------------------------------- # restrict reference configurations #--------------------------------------- RejectInvalidRefs true # by default reference CSFs are restricted # to target spin and spatial symmetry # ====================================== # Definitions of blocks of the CI Matrix # ====================================== NewBlock 2 * # generate a Block with doublet(=2) multiplicity Nroots 1 # number of roots to be generated Excitations cis # CI with single excitations cid # CI with double excitations cisd # CI with single and double excitations ddci1 # DDCI list with one degree of freedom ddci2 # DDCI list with two degrees of freedom ddci3 # DDCI list with three degrees of freedom Flags[_class_] 0 or 1 # Turn excitation classes on or off individually # ``s'' stands for any SOMO, ``i'',``j'' for internal orbitals and # ``a'',``b'' for external orbitals # Singles _class_ = ss, sa, is, ia # Doubles _class_ = ijss, ijsa, ijab, # isss, issa, isab, # ssss, sssa, ssab # ``Flags'' takes priority over ``Excitations''. In fact ``Excitations'' # does nothing but to set ``Flags''. So, you can use ``Excitations'' # to provide initial values for ``Flags'' and then modify them # with subsequent ``Flags'' assignments refs # # First choice - complete active space # CAS(nel,norb) # CAS-CI reference with nel electrons in # Norb orbitals # # Second choice - restricted active space # RAS(nel: m1 h/ m2 / m3 p) # RAS-reference with nel electrons # m1= number orbitals in RAS-1 # h = max. number of holes in RAS-1 # m2= number of orbitals in RAS-2 (any number of # electrons or holes) # m3= number of orbitals in RAS-3 # p = max. number of particles in RAS-3 # # Third choice - individually defined configurations # \{ 2 0 1 0\} \{ 1 1 1 0\} etc. # define as many configurations as you want. Doubly occupied MOs # singly occupied MOs and empty MOs. Important notes: # a) the number of electrons must be the same in all references # b) the number of orbitals is determined from the number of # definitions. Thus, in the example above we have three active # electrons and four active orbitals despite the fact that the # highest orbital is not occupied in any reference. # The program determines the internal, active and external spaces # automatically from the number of active electrons and orbitals end end # there can be as many blocks as you want!!! # ---------------------------------- # Density matrix generation flags # First Key= State densities # =0: none # =1: Ground state only (lowest root of all blocks; Electron only) # =2: Ground state only (Electron and spin density) # =3: Lowest root from each block (Electron density) # =4: Lowest root from each block (Electron and spin density) # =5: All states (Electron density) # =6: All states (Electron and spin density) # Second Key= Transition densities # needed for all transition intensities, g-tensor etc # =0: none # =1: from the ground state into all excited states (el) # =2: from the ground state into all excited states (el+spin) # =3: from all lowest states into all excited states (el) # =4: from all lowest states into all excited states (el+spin) # =5: all state pairs (el) # =6: all state pairs (el+spin) # Note that for perturbation theory the density is computed as # an expectation value over the first (second) order wavefunction. # which is renormalized for this purpose # ---------------------------------- Densities 1,1 # ---------------------------------- # Complete printing of the wavefunction # ---------------------------------- PrintWF 1 # CFG based printing (default) det # Determinant based wavefunction printing TPrintWF 3e-3 # Threshold for the printing of the CFGs/Dets # ---------------------------------- # Algorithm for the solver # ---------------------------------- Solver Diag # Davidson like solver DIIS # DIIS like solver # both solvers have their pros and cons. The DIIS may converge # better or use less time since it only recomputes the vectors that # have not yet converged; The DIIS may be less sensitive to root flipping # effects but occasionally it converges poorly and states of the same # symmetry are occasionally a little problematic # For perturbation theory DIIS is always used. # For both solvers MaxIter 100 # the maximum number of iterations Etol 1e-6 # convergence tolerance for energies in Eh Rtol 1e-6 # convergence tolerance for residual # For Solver=Diag (Davidson solver) Otol 1e-16 # Orthogonality threshold for Schmidt process NGuessMat 512 # Dimension of the guess matrix 512x512 # be used to compute the initial guess of the actual MRCI calculation NGuessMatRefCI 512 # Dimension of the guess matrix # for the reference CI MaxDim 3 # Davidson expansion space = MaxDim * NRoots # For the Solver=DIIS. Particularly recommended for anything else but # straightforward CI and also for calculations in direct2 mode! MaxDIIS 5 # Maximum number of old vectors to be used in DIIS RelaxRefs true # Relax reference space coefficients in the CI or # freeze them to their zeroth order values LevelShift 0.4 # Level Shift for stabilizing the DIIS procedure # ---------------------------------- # RI Approximation # ---------------------------------- IntMode RITrafo #Use RI integrals FullTrafo #No RI (default) # ---------------------------------- # Integral storage, memory and files # ---------------------------------- IntStorage FloatVals DoubleVals (default) # store integrals with float (4 byte) or double (8 byte) # accuracy in main memory FourIndexInts false (default) True # Store ALL four index integrals over Mos in main memory # only possible for relatively small systems, perhaps up # to 150-200 MOs included in the CI MaxMemInt 256 # Maximum amount of core memory devoted to the storage of # integrals. If NOT all three index integrals fit into main # memory the program fails MaxMemVec 16 # Maximum amount of memory used for section of the trial and # sigma vectors. This is not a particularly critical variable KeepFiles false # Keep integrals and CI program input file (.mrciinp). Then # you can manually edit the .mrciinp file which is a standard # ASCII file and run the MRCI program directly. The only thing # you cannot change is the orbital window. end ``` (sec:mrci.soc.detailed)= ## Properties Calculation Using the SOC Submodule (sec:mrci.soc.zfs.detailed)= ### Zero-Field Splitting The spin-orbit coupling (SOC) and spin-spin coupling (SSC) contributions to the zero-field splitting (ZFS) can be calculated very accurately using a wavefunction obtained from a multiconfigurational calculation of a multi-reference type such as CASSCF, MRCI, or MRPT as is described in QDPT Magnetic Properties Section {ref}`sec:qdpt_magnetic_properties.detailed`. ``` # In case that you want to run QDPT-SOC calculation with manually #adjusted diagonal energies you can copy the following part into #the %mrci soc block #and modify it as needed(energies are given in #wavenumbers relative to the lowest state) # NOTE: It is YOUR responsibility to make sure that the CAS-CI state #that you may want to dress with these energies correlate properly #with the energies printed here. The order of states or even the #identity of states may change with and without inclusion of #dynamic correlation In the case that dynamic correlation strongly #mixes different CAS-CI states there may not even be a proper #correlation! # EDiag[ 0] 0.00 # root 0 of block 0 EDiag[ 1] 48328.40 # root 1 of block 0 EDiag[ 2] 48328.40 # root 2 of block 0 EDiag[ 3] 49334.96 # root 3 of block 0 EDiag[ 4] 7763.59 # root 0 of block 1 EDiag[ 5] 7763.59 # root 1 of block 1 EDiag[ 6] 11898.46 # root 2 of block 1 EDiag[ 7] 46754.23 # root 3 of block 1 ``` Those transition energies can be substituted by a more accurate energies provided in the input file as follows: ``` %soc dosoc true dossc true EDiag[ 0] 0.00 # root 0 of block 0 EDiag[ 1] 48328.40 # root 1 of block 0 EDiag[ 2] 48328.40 # root 2 of block 0 EDiag[ 3] 49334.96 # root 3 of block 0 EDiag[ 4] 7763.59 # root 0 of block 1 EDiag[ 5] 7763.59 # root 1 of block 1 EDiag[ 6] 11898.46 # root 2 of block 1 EDiag[ 7] 46754.23 # root 3 of block 1 end ``` Accurate diagonal energies generally improve the accuracy of the SOC and SSC splittings. (sec:mrci.soc.lzfs.detailed)= ### Local Zero-Field Splitting The submodule can also be used to calculate the local ZFS splitting parameters of atomic centers. The method, referred to as local complete active space configuration interaction (L-CASCI), can be used to separate into atomic contributions the SOC part of the total ZFS tensor. The rational behind it and additional details are described in the original publication {cite}`retegan_first-principles_2014`; below are listed only the steps required to reproduce the calculation for the dimer complex presented there. 1\. The first step consists in obtaining the molecular orbitals that are going to be used in the configuration interaction (CI) procedure. A good set of orbitals can be obtained from a restricted open-shell spin-averaged Hartree-Fock (SAHF) calculation. The relevant part of the input is listed below: ```orca ! def2-tzvp keepfock % scf hftyp rohf rohf_case sahf rohf_numop 2 rohf_nel[1] 9 rohf_norb[1] 10 end ``` For the present Mn(II)Mn(III) dimer there are a total of 9 electrons distributed into 10 d-orbitals. 2\. Next, the molecular orbitals are localized using one of the implemented localization schemes. Below is the `orca_loc` input used in this case: ```orca sahf.gbw sahf.loc 0 200 # first of the 10 d-orbitals 209 # last of the 10 d-orbitals 128 0.000001 0.75 0.65 2 ``` 3\. Following this, the localized orbitals are made locally canonical by block diagonalizing the Fock matrix using the `orca_blockf` utility. ```orca orca_blockf sahf.fsv sahf.loc 200 204 205 209 ``` The first two numbers define the range of molecular orbitals localized on one center; the last two are for the second center. 4\. The recanonicalized orbitals stored in the `sahf.loc` file can be then used to calculate the SOC contribution to the local ZFS of the Mn(III) center using the following MRCI input: ```orca ! zora-def2-tzvp def2-tzvp/c zora ! nomulliken noloewdin ! moread noiter allowrhf ! moread % mrci citype mrci tsel 0 tpre 0 intmode ritrafo solver diis soc intmode ritrafo dosoc true end newblock 10 * nroots 5 excitations none refs # Mn(II) Mn(III) {1 1 1 1 1 1 1 1 1 0} {1 1 1 1 1 1 1 1 0 1} {1 1 1 1 1 1 1 0 1 1} {1 1 1 1 1 1 0 1 1 1} {1 1 1 1 1 0 1 1 1 1} end end newblock 8 * nroots 45 excitations none refs # Mn(II) Mn(III) {1 1 1 1 1 2 1 1 0 0} {1 1 1 1 1 2 1 0 1 0} {1 1 1 1 1 2 1 0 0 1} {1 1 1 1 1 2 0 1 1 0} {1 1 1 1 1 2 0 1 0 1} {1 1 1 1 1 2 0 0 1 1} {1 1 1 1 1 1 2 1 0 0} {1 1 1 1 1 1 2 0 1 0} {1 1 1 1 1 1 2 0 0 1} {1 1 1 1 1 1 1 2 0 0} {1 1 1 1 1 1 1 1 1 0} {1 1 1 1 1 1 1 1 0 1} {1 1 1 1 1 1 1 0 2 0} {1 1 1 1 1 1 1 0 1 1} {1 1 1 1 1 1 1 0 0 2} {1 1 1 1 1 1 0 2 1 0} {1 1 1 1 1 1 0 2 0 1} {1 1 1 1 1 1 0 1 2 0} {1 1 1 1 1 1 0 1 1 1} {1 1 1 1 1 1 0 1 0 2} {1 1 1 1 1 1 0 0 2 1} {1 1 1 1 1 1 0 0 1 2} {1 1 1 1 1 0 2 1 1 0} {1 1 1 1 1 0 2 1 0 1} {1 1 1 1 1 0 2 0 1 1} {1 1 1 1 1 0 1 2 1 0} {1 1 1 1 1 0 1 2 0 1} {1 1 1 1 1 0 1 1 2 0} {1 1 1 1 1 0 1 1 1 1} {1 1 1 1 1 0 1 1 0 2} {1 1 1 1 1 0 1 0 2 1} {1 1 1 1 1 0 1 0 1 2} {1 1 1 1 1 0 0 2 1 1} {1 1 1 1 1 0 0 1 2 1} {1 1 1 1 1 0 0 1 1 2} end end end ``` 5\. The three second order ZFS components printed at the end of the calculation (`Second order D-tensor: component 0`, etc.) are scaled using the *S* value for the complex, which in this case is 4.5 (9 electrons $\times$ 0.5). In order to obtain the correct local value of the ZFS, the three matrices have to be rescaled using the *S* value for Mn(III), which is to 2. Note that the three matrices have different scaling prefactors, and the dependence on *S* is not the same: $\mathbf{D}^{SOC-(0) } \propto \frac{1}{S^2}$ $\mathbf{D}^{SOC-(-1) } \propto \frac{1}{S(2S-1) }$ $\mathbf{D}^{SOC-(+1) } \propto \frac{1}{(S+1)(2S+1) }$ These equations can be used to calculate the required prefactors. For example in the case of the *SOC*-(0) the prefactor is equal to: $\mathbf{D}_{\mathrm{Mn(III) }}^{SOC-(0) } = \frac{4.5^2}{2^2}\cdot\mathbf{D}_{\mathrm{dimer} }^{SOC-(0) } = 5.0625 \cdot \mathbf{D}_{\mathrm{dimer} }^{SOC-(0) }$ The final step is to scale the two remaining matrices using the appropriate prefactors, sum all three of them up, diagonalize the resulting the matrix, and use its eigenvalues to calculate the *D* and *E* parameters. These represent the local ZFS parameters of the Mn(III) center. (sec:mrci.soc.excited_state_zfs.detailed)= ### Zero-Field Splitting from an excited Multiplet For an excited state Multiplet the Calculationof ZFS can be requested by ```orca 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 ``` ```orca soc DTensor true IStates 3,4,5 end ``` ```orca ***************************************** EXCITED STATE ZERO-FIELD SPLITTING: ***************************************** -------------------------------------------- Computing Excited State D Tensors of Excited State Multiplet Consisting of States : 3 4 5 -------------------------------------------- 0 4 1 5 2 0 3 1 4 0 5 2 -------------------------------------------- ZERO-FIELD SPLITTING (2ND ORDER SPIN-ORBIT COUPLING CONTRIBUTION) -------------------------------------------- D = -2.668445 cm-1 E/D = 0.000103 ... -------------------------------------------------------- ZERO-FIELD SPLITTING EFFECTIVE HAMILTONIAN SOC CONTRIBUTION -------------------------------------------------------- D = -2.674495 cm-1 E/D = 0.009610 ... ``` (sec:mrci.soc.gTensor.detailed)= ### g-Tensor The `orca_mrci` program contains an option to calculate g-tensors using MRCI wavefunctions. For a system with an odd number of electrons, the doubly degenerate eigenvalues obtained from the QDPT procedure represent Kramers pairs, which are used to build the matrix elements of the total spin operator and the total angular momentum operator from the Zeeman Hamiltonian. Denoting $\Psi$ as a solution and $\bar{\Psi}$ as its Kramers partner and using matrix element notations $$\Phi_{11}^{k} =\left\langle\Psi \right|\hat{{L} }_{k} +g_{e} \hat{{S} }_{k} \left| \Psi \right\rangle,\, \Phi_{12}^{k} =\left\langle\Psi \right|\hat{{L} }_{k} +g_{e} \hat{{S} }_{k} \left| \bar{\Psi} \right\rangle,\, k=x,y,z $$ (eqn:229) The elements of g-matrix are obtained as: $$g_{kz} =2\Phi_{11}^{k} ,\, g_{ky} =-2\Im\left({ \Phi_{12}^{k} } \right) ,\, g_{kx} =2\Re\left({ \Phi_{12}^{k} } \right)$$ (eqn:230) Then, the true tensor G is built from g-matrices: $$G=gg^{T} $$ (eqn:231) G is subjected further to diagonalization yielding positive eigenvalues, the square roots of which give the principal values of g-matrix. $$g_{xx} =\sqrt{ G_{xx} } ,\, g_{yy} =\sqrt{ G_{yy} } ,\, g_{zz} =\sqrt{ G_{zz} } $$ (eqn:232) A typical mrci block of the input file for a g-tensor calculation should (e.g. for a S=3/2 problem) look as the following: ```orca %mrci ewin -4,1000 citype mrci cimode direct2 intmode fulltrafo solver diis etol 1e-8 rtol 1e-8 tsel 1e-6 tpre 1e-5 soc PrintLevel 2 GTensor true # make g-tensor calculations NDoubGTensor 2 # number of Kramers doublets to account # for every pair a separate # calculation is performed end newblock 4 * excitations cisd nroots 10 refs cas(7,5) end end end ``` The result for the first Kramers pair is printed as follows: ``` -------------- KRAMERS PAIR 1 -------------- Matrix elements Re<1|S|1> -0.072128 0.024511 -2.998843 Matrix elements Re<1|S|2> -0.001088 0.000366 -0.002010 Matrix elements Im<1|S|2> -0.000354 -0.001037 -0.000173 Matrix elements Re<1|L|1> -0.027067 0.009209 -1.123531 Matrix elements Re<1|L|2> -0.000031 0.000010 -0.000763 Matrix elements Im<1|L|2> -0.000006 -0.000011 -0.000065 ------------------- ELECTRONIC G-MATRIX ------------------- g-matrix: -0.002240 0.000754 -0.005551 0.000720 0.002100 0.000477 -0.198556 0.067498 -8.251703 g-factors: 0.002220 0.002222 8.254370 iso = 2.752937 g-shifts: -2.000100 -2.000098 6.252051 iso = 0.750618 Eigenvectors: 0.057426 0.998060 0.024055 0.998327 -0.057244 -0.008177 0.006784 -0.024484 0.999677 ``` Here for the $L$ and $S$ matrix elements indices 1 and 2 are assumed to denote Kramers partners, and three numbers in the first row stand for $x, y, z$ contributions. In addition the g-tensor is calculated within the Effective Hamiltonian formalism. ``` ---------------------------------------------- ELECTRONIC G-MATRIX FROM EFFECTIVE HAMILTONIAN ---------------------------------------------- g-matrix: 1.978874 -0.000345 0.018908 -0.000345 1.977899 -0.006433 0.018879 -0.006418 2.763402 g-factors: 1.977789 1.978477 2.763909 iso = 2.240058 g-shifts: -0.024530 -0.023843 0.761590 iso = 0.237739 Eigenvectors: 0.288884 0.957062 0.024060 0.957364 -0.288770 -0.008181 0.000882 -0.025397 0.999677 # The g-factors are square roots of the eigenvalues of gT*g # Orientations are the eigenvectors of gT*g ``` Finally and only within the MRCI module the g-tensor is evaluated by using the Sum Over States formalism{cite}`neese2003sos-gtensors`: ``` --------------------------------------------------------------------------- SUM OVER STATES CALCULATION OF THE SPIN HAMILTONIAN (for g and HFC tensors) --------------------------------------------------------------------------- Ground state index = 0 Ground state multiplicity = 4 Ground state spin density = P[ 1] State = 1 <0|P|I>= 2 <0|Q|I>= 19 State = 2 <0|P|I>= 3 <0|Q|I>= 27 State = 3 <0|P|I>= 4 <0|Q|I>= 34 State = 4 <0|P|I>= 5 <0|Q|I>= 40 State = 5 <0|P|I>= 6 <0|Q|I>= 45 State = 6 <0|P|I>= 7 <0|Q|I>= 49 State = 7 <0|P|I>= 8 <0|Q|I>= 52 State = 8 <0|P|I>= 9 <0|Q|I>= 54 State = 9 <0|P|I>= 10 <0|Q|I>= 55 Origin for angular momentum ... ( -0.0006, -0.0010, 0.0021) Kinetic Energy ... done Relativistic mass correction ... done Gauge correction ... done Angular momentum integrals ... done Reading Spin-Orbit Integrals ... done ----------------------- MATRIX ELEMENT PRINTING ----------------------- Energy differences (DE=EI-E0) and spin-orbit matrix elements (SO=) are printed in cm**-1. Orbital Zeeman matrix elements (L=) are printed in au. State DE LX LY LZ SOX SOY SOZ 1 1349.3 0.0464 -0.0158 1.9264 -23.432 7.965 -974.312 2 13026.2 -0.6596 0.6888 0.0214 337.028 -351.116 -10.966 3 13615.1 -0.6961 -0.6514 0.0113 354.225 332.219 -5.736 4 56686.3 -0.0053 0.0077 0.0971 1.794 -1.696 -36.786 5 56954.2 -0.0516 -0.0048 -0.0042 28.211 5.821 1.459 6 56994.0 -0.0418 0.0233 -0.0025 15.185 -2.144 1.145 7 63371.5 -0.0211 0.0226 0.0078 3.833 -2.948 -2.724 8 64176.0 -0.0652 0.0032 -0.0002 32.779 6.146 0.063 9 74309.9 -0.0007 0.0032 0.0380 0.183 -1.058 -13.517 ------------------- ELECTRONIC G-MATRIX ------------------- raw-matrix : 2.025533 -0.000738 0.021755 -0.000738 2.024537 -0.007389 0.021755 -0.007389 2.928943 g-factors: 2.024122 2.025363 2.929527 iso = 2.326338 g-shifts: 0.021803 0.023044 0.927208 iso = 0.324018 Eigenvectors: 0.533896 -0.845208 0.024064 0.845530 0.533866 -0.008182 -0.005932 0.024715 0.999677 Euler angles w.r.t. molecular frame (degrees): -76.5038 1.4564 -161.2223 ----------------------------- CONTRIBUTIONS TO THE G-MATRIX ----------------------------- Term g1 g2 g3 -------------------------------------------------------------------- Relativistic mass correction: -0.0008220 -0.0008220 -0.0008220 Gauge correction : 0.0000000 0.0000000 0.0000000 g(OZ/SOC) : 0.0226250 0.0238662 0.9280297 State 1 : 0.0000000 -0.0000000 0.9279829 State 2 : 0.0013767 0.0223913 0.0000000 State 3 : 0.0212332 0.0014408 0.0000000 State 4 : 0.0000000 0.0000004 0.0000418 State 5 : 0.0000074 0.0000099 0.0000001 State 6 : 0.0000002 0.0000078 0.0000001 State 7 : 0.0000000 0.0000015 0.0000002 State 8 : 0.0000076 0.0000144 0.0000000 State 9 : 0.0000000 0.0000000 0.0000046 ----------------------------------------- Total g-shifts : 0.0218030 0.0230442 0.9272077 # The g-factors are square roots of the eigenvalues of gT*g # Orientations are the eigenvectors of gT*g ``` Note that within the SOS formalism in addition to the second order (SOC) contributions the bilinear to the field terms: Relativistic mass correction and diamagnetic spin-orbit term (Gauge) are evaluated. As can be seen these corrections are rather negligible in comparison to the second order SOC contributions and most of the time can be safely omitted. Moreover further insight is obtained by printing the individual contribution of each excited state to the g-tensor. In the example above the first excited state contributes to the $g_z$ component while the next two to both the $g_x$ and $g_y$ components, respectively. So to summarize the g-tensor calculations in the framework of wavefunction based methods like MRCI and/or CASSCF can be evaluated: - via the QDPT approach within an individual Kramers doublet. This is valid analysis only for non-integer spin cases. In particular for systems with well isolated Kramers doublets where the EPR spectrum originates only from one Kramers doublet defined within the pseudo spin 1/2 formalism. This analysis has been proven useful in determining the sign of the ZFS and the electronic structure of the system under investigation.{cite}`maganas2011zfs` - within the effective Hamiltonian approach. This is a valid analysis for all spin cases as it provides the principal g-values of the system under investigation evaluated in the molecular axis frame. These g-values can be directly compared with the experimentally determined ones.{cite}`maganas2015zfs` - within the sum over states formalism (SOS). As above this analysis is valid for all spin cases and is only available via the MRCI module. (sec:mrci.soc.magnet.detailed)= ### Magnetization and Magnetic Susceptibility The MRCI and CASSCF modules of ORCA allow for the calculation of magnetization and magnetic susceptibility curves at different fields and temperatures by differentiation of the QDPT Hamiltonian with respect to the magnetic field. For magnetic susceptibility, calculations are performed in two ways when a static field different from zero is defined: (i) as the second derivative of energy with respect to the magnetic field and (ii) as the magnetization divided by the magnetic field. Although the first method corresponds to the definition of magnetic susceptibility, the second approach is widely used in the experimental determination of $\chi*T$ curves. If the static field is low, both formulas tend to provide similar values. The full list of keywords is presented below. ```orca %mrci citype mrci newblock 3 * excitations none refs cas(2,7) end end soc dosoc true domagnetization true # Calculate magnetization (def: false) dosusceptibility true # Calculate susceptiblity (def: false) LebedevPrec 5 # Precision of the grid for different field # directions (meaningful values range from 1 # (smallest) to 10 (largest)) nPointsFStep 5 # number of steps for numerical differentiation # (def: 5, meaningful values are 3, 5 7 and 9) MAGFieldStep 100.0 # Size of field step for numerical differentiation # (def: 100 Gauss) MAGTemperatureMIN 4.0 # minimum temperature (K) for magnetization MAGTemperatureMAX 4.0 # maximum temperature (K) for magnetization MAGTemperatureNPoints 1 # number of temperature points for magnetization MAGFieldMIN 0.0 # minimum field (Gauss) for magnetization MAGFieldMAX 70000.0 # maximum field (Gauss) for magnetization MAGNpoints 15 # number of field points for magnetization SUSTempMIN 1.0 # minimum temperature (K) for susceptibility SUSTempMAX 300.0 # maximum temperature (K) for susceptibility SUSNPoints 300 # number of temperature points for susceptibility SUSStatFieldMIN 0.0 # minimum static field (Gauss) for susceptibility SUSStatFieldMAX 0.0 # maximum static field (Gauss) for susceptibility SUSStatFieldNPoints 1 # number of static fields for susceptibility end end ``` The same keywords apply for CASSCF calculations in rel block (instead of soc in MRCI). Although different aspects of integration and grid precision can be modified through keywords, default values should provide an accurate description of both properties. Calculated magnetization and susceptibility are printed in .sus and .mag files, respectively and also in the output file. ``` ------------------------------------------------------------------------------- FIELD DEPENDENT MAGNETIZATION AND MEAN SUSCEPTIBILITY (chi=M/B) ------------------------------------------------------------------------------- TEMPERATURE (K) M. FIELD (Gauss) MAGNETIZATION (B.M.) chi*T (cm3*K/mol) ------------------------------------------------------------------------------- 4.00 0.00 0.000000 inf 4.00 5000.00 0.350759 1.567189 4.00 10000.00 0.688804 1.538788 4.00 15000.00 1.003466 1.494496 4.00 20000.00 1.287480 1.438115 4.00 25000.00 1.537346 1.373773 4.00 30000.00 1.752841 1.305282 4.00 35000.00 1.936067 1.235764 4.00 40000.00 2.090450 1.167516 4.00 45000.00 2.219920 1.102067 4.00 50000.00 2.328368 1.040315 4.00 55000.00 2.419335 0.982690 4.00 60000.00 2.495883 0.929301 4.00 65000.00 2.560582 0.880052 4.00 70000.00 2.615538 0.834730 ----------------------------------------------------------- ``` ``` ----------------------------------------------------------- TEMPERATURE DEPENDENT MAGNETIC SUSCEPTIBILITY ----------------------------------------------------------- STATIC FIELD TEMPERATURE chi*T (cm3*K/mol) (Gauss) (K) M/B d2E/dB2 ----------------------------------------------------------- 0.00 1.00 ---- 1.576836 0.00 2.00 ---- 1.576910 0.00 3.00 ---- 1.576951 0.00 4.00 ---- 1.576988 0.00 5.00 ---- 1.577023 0.00 6.00 ---- 1.577057 0.00 7.00 ---- 1.577091 0.00 8.00 ---- 1.577125 0.00 9.00 ---- 1.577159 0.00 10.00 ---- 1.577193 0.00 11.00 ---- 1.577227 ..... 0.00 300.00 ---- 1.586942 1000.00 1.00 1.570517 1.558042 1000.00 2.00 1.575324 1.572178 1000.00 3.00 1.576246 1.574845 1000.00 4.00 1.576590 1.575802 1000.00 5.00 1.576768 1.576264 1000.00 6.00 1.576880 1.576530 1000.00 7.00 1.576961 1.576704 1000.00 8.00 1.577026 1.576829 ..... ``` Note that the CASSCF module also supports the calculation of susceptibility tensors at non-zero user-defined magnetic fields. This is not yet possible with the MRCI module. (sec:mrci.soc.mcd.detailed)= ### MCD and Absorption Spectra The MRCI module of the ORCA program allows calculating MCD spectra and the SOC effects on absorption spectra. The formalism is described in detail by Ganyushin and Neese{cite}`ganyushin2008`. The approach is based on the direct calculation of the transition energies and transition probabilities between the magnetic levels. Namely, the differential absorption of LCP- and RCP photons for transitions from a manifold of initial states $A$ to a manifold of final states $J$. Using Fermi's golden rule, the Franck-Condon approximation, assuming a pure electronic dipole mechanism and accounting for the Boltzmann populations of the energy levels, the basic equation of MCD spectroscopy may be written as (atomic units are used throughout): $$\frac{\Delta \varepsilon }{E}=\gamma \sum\limits_{a,j} { \left({N_{a} -N_{j} } \right)\left({ \left|{ \left\langle { \Psi_{a} \left|{m_{\text{LCP} } } \right|\Psi_{j} } \right\rangle} \right|^{2}-\left|{\left\langle { \Psi_{a} \left|{ m_{\text{RCP} } } \right|\Psi_{j} } \right\rangle} \right|^{2} } \right)f\left( E \right)} $$ (eqn:233) Here $a$ and $j$ label members of the initial and state manifold probed in the experiments. $$N_{a} \left({ B,T} \right)=\frac{\exp \left({ -E_{a} /kT} \right)}{\sum\limits_i { \exp \left({ -E_{i} /kT} \right)} } $$ (eqn:234) denotes the Boltzmann population and if the $a$-th ground state sublevel at energy $E_{a}$, $f\left( E \right)$ stands for a line shape function, and $\gamma$ denotes a collection of constants. The electric dipole operators are given by: $$m_{\text{LCP} } \equiv m_{x} -im_{y} $$ (eqn:235) $$m_{\text{RCP} } \equiv m_{x} +im_{y} $$ (eqn:236) They represent linear combinations of the dipole moment operator: $$\vec{{m} }=\sum\limits_N { Z_{N} \vec{{R} }_{N} } -\sum\limits_i { \vec{{r} }_{i} } $$ (eqn:237) where $N$ and $i$ denotes summations of nuclei (at positions $\vec{{R} }_{N}$ with charges $Z_{N})$ and electrons (at positions $\vec{{r} }_{i} )$ respectively. The calculated transition dipole moment are subjected to the space averaging over the Euler angles which is performed by a simple summation over three angular grids. $$\left({ \frac{\Delta \varepsilon }{E} } \right)_{ev} =\frac{1}{8\pi ^{2} }\int\limits_{\psi =0}^{2\pi } { \int\limits_{\phi =0}^{2\pi } {\int\limits_{\theta =0}^\pi{ \left({ \frac{\Delta \varepsilon }{E} } \right)\sin \theta d\theta d\phi d\psi } } } \approx \sum\limits_{\mu \eta \tau } { \left({ \frac{\Delta \varepsilon }{E} } \right)_{\mu \eta \tau } } \sin \theta_{\tau } $$ (eqn:238) Finally, every transition is approximated by a Gaussian curve with a definite Gaussian shape width parameter. Hence, the final calculated MCD spectrum arises from the superposition of these curves. As an illustration, consider calculation of a classical example of MCD spectrum of \[Fe(CN) $_{6}$\]$^{3-}$. The mrci block of the input file is presented below. ```orca %mrci ewin -4,10000 citype mrddci2 intmode ritrafo Tsel 1e-6 Tpre 1e-5 etol 1e-8 rtol 1e-8 cimode direct2 maxmemint 300 solver diis davidsonopt 0 nguessmat 150 MaxIter 50 LevelShift 0.5 PrintLevel 3 soc printlevel 3 Domcd true # perform the MCD calculation NInitStates 24 # number of SOC and SSC state to account # Starts from the lowest state NPointsTheta 10 # number of integration point for NPointsPhi 10 # Euler angles NPointsPsi 10 # B 43500 # experimental magnetic field strength # in Gauss Temperature 299.0 # experimental temperature (in K) end newblock 2 * nroots 12 excitations cisd refs cas(23,12) end end end ``` The parameters B and Temperature can be assigned in pairs, i.e. B $=$ 1000, 2000, 3000..., Temperature $=$ 4, 10, 300.... The program calculates the MCD and absorption spectra for every pair. Now for every point of the integration grid the program prints out the Euler angles, the orientation of the magnetic field in the coordinate system of a molecule, and the energy levels. ```orca Psi = 36.000 Phi = 72.000 Theta = 20.000 Bx = 8745.0 By = 12036.5 Bz = 40876.6 Energy levels (cm-1,eV):Boltzmann populations for T = 299.000 K 0 : 0.000 0.0000 4.53e-01 1 : 3.943 0.0005 4.45e-01 2 : 454.228 0.0563 5.09e-02 3 : 454.745 0.0564 5.08e-02 4 : 1592.142 0.1974 2.13e-04 5 : 1595.272 0.1978 2.10e-04 6 : 25956.363 3.2182 2.59e-55 7 : 25958.427 3.2184 2.56e-55 8 : 25985.656 3.2218 2.25e-55 9 : 25987.277 3.2220 2.23e-55 10 : 26070.268 3.2323 1.49e-55 11 : 26071.484 3.2325 1.49e-55 12 : 31976.645 3.9646 6.78e-68 13 : 31979.948 3.9650 6.67e-68 14 : 32018.008 3.9697 5.56e-68 15 : 32021.074 3.9701 5.48e-68 16 : 32153.427 3.9865 2.90e-68 17 : 32157.233 3.9870 2.84e-68 18 : 42299.325 5.2444 1.81e-89 19 : 42303.461 5.2450 1.78e-89 20 : 42346.521 5.2503 1.45e-89 21 : 42348.023 5.2505 1.44e-89 22 : 42456.119 5.2639 8.53e-90 23 : 42456.642 5.2640 8.51e-90 ``` In the next lines, ORCA calculates the strength of LCP and RCP transitions and prints the transition energies, the difference between LCP and RCP transitions (denoted as C), and sum of LCP and RCP transitions (denoted as D), and C by D ratio. ```orca dE Na C D C/D 0 -> 1 3.943 4.53e-01 1.14e-13 8.13e-13 0.00e+00 0 -> 2 454.228 4.53e-01 5.01e-09 9.90e-09 5.06e-01 0 -> 3 454.745 4.53e-01 -4.65e-09 7.00e-09 -6.65e-01 0 -> 4 1592.142 4.53e-01 -8.80e-08 1.02e-07 -8.67e-01 0 -> 5 1595.272 4.53e-01 -2.29e-08 2.97e-08 -7.71e-01 0 -> 6 25956.363 4.53e-01 1.22e+01 9.60e+01 1.27e-01 0 -> 7 25958.427 4.53e-01 3.44e+01 3.52e+01 9.77e-01 0 -> 8 25985.656 4.53e-01 3.83e+01 1.70e+02 2.25e-01 0 -> 9 25987.277 4.53e-01 -7.73e+00 6.03e+01 -1.28e-01 0 ->10 26070.268 4.53e-01 -6.11e+00 2.85e+01 -2.14e-01 0 ->11 26071.484 4.53e-01 6.17e+00 9.21e+00 6.70e-01 0 ->12 31976.645 4.53e-01 2.45e+01 6.21e+01 3.95e-01 0 ->13 31979.948 4.53e-01 -6.58e+01 6.93e+01 -9.50e-01 0 ->14 32018.008 4.53e-01 3.42e-01 1.07e+02 3.21e-03 0 ->15 32021.074 4.53e-01 -6.16e+00 3.24e+01 -1.90e-01 0 ->16 32153.427 4.53e-01 -4.73e+01 1.37e+02 -3.46e-01 0 ->17 32157.233 4.53e-01 -1.02e+00 5.97e+01 -1.71e-02 0 ->18 42299.325 4.53e-01 6.47e+00 2.11e+01 3.07e-01 0 ->19 42303.461 4.53e-01 -2.59e+00 7.61e+00 -3.40e-01 0 ->20 42346.521 4.53e-01 1.90e+01 8.99e+01 2.11e-01 0 ->21 42348.023 4.53e-01 3.36e+00 3.55e+00 9.48e-01 0 ->22 42456.119 4.53e-01 2.52e-01 4.86e-01 5.20e-01 0 ->23 42456.642 4.53e-01 -2.01e+00 2.91e+00 -6.91e-01 1 -> 2 450.285 4.45e-01 4.59e-09 6.87e-09 6.69e-01 1 -> 3 450.802 4.45e-01 -4.96e-09 9.73e-09 -5.09e-01 ``` All C and D values are copied additionally into the text files input.1.mcd, input.2.mcd..., for every pair of Temperature and B parameters. These files contain the energies and C and D values for every calculated transition. These files are used by the program `orca_mapspc` to calculate the spectra lines. The `orca_mapspc` program generates from the raw transitions data into spectra lines. The main parameters of the `orca_mapspc` program are described in section 7.18.1. A typical usage of the `orca_mapspc` program for MCD spectra calculation for the current example may look as the following: ```orca orca_mapspc input.1.mcd MCD -x020000 -x150000 -w2000 ``` Here the interval for the spectra generation is set from 20000 cm$^{-1}$ to 50000 cm$^{-1}$, and the line shape parameter is set to 2000 cm$^{-1}$. Very often, it is desirable to assign different line width parameters to different peaks of the spectra to obtain a better fitting to experiment. `orca_mapspc` can read the line shape parameters from a simple text file named as input.1.mcd.inp. This file should contain the energy intervals (in cm$^{-1})$ and the line shape parameters for this energy interval in the form of: ```orca 20000 35000 1000 35000 40000 2000 40000 50000 1000 ``` This file should not be specified in the executing command; `orca_mapspc` checks for its presence automatically: ```orca orca_mapspc input.1.mcd MCD -x020000 -x150000 Mode is MCD Number of peaks ... 276001 Start wavenumber [cm-1] ... 20000.0 Stop wavenumber [cm-1] ... 50000.0 Line width parameters are taken from the file:input.1.mcd.inp Number of points ... 1024 ``` Finally, the `orca_mapspc` program generates the output text file input.1.mcd.dat which contains seven columns of numbers: transition energies, intensities of MCD transitions (the MCD spectrum), intensities of absorption transitions (the absorption spectrum), the ratio between the MCD and absorption intensities, and the last three columns represent the "sticks" of the corresponding transitions. ```orca Energy C D C/D C D E/D 24310.8 0.6673 980.2678 0.0006 0.0000 0.0000 0.0000 24340.1 0.8471 1174.3637 0.0007 -0.0001 0.0129 -0.0112 24369.5 1.0664 1408.5788 0.0007 0.0001 0.0281 0.0033 24398.8 1.3325 1690.5275 0.0007 0.0000 0.0000 0.0000 24428.1 1.6542 2029.0152 0.0008 0.0000 0.0000 0.0000 24457.4 2.0416 2434.1699 0.0008 0.0000 0.0332 0.0003 ``` Now the MCD and the absorption spectra can be plotted with a suitable graphical program, for instance with the Origin program. (fig:713)= ```{figure} ../../images/713.* Calculated MCD and absorption spectra of [Fe(CN) $_6$]$^{3-}$ (dash lines) compared to experimental spectra (solid lines). ``` (sec:mrci.soc.magField.detailed)= ### Addition of Magnetic Fields The inclusion of the Zeeman contribution into the QDPT procedure allows to obtain the splittings of the magnetic levels in an external magnetic field. The switch for this calculation and the magnetic field strength are defined in the soc subblock of the mrci block. Optionally the wave function decomposition can be printed for `MagneticField_PrintLevel` larger 0. The latter employs the thresh `TPrint` to omit small contributions from the printing: ```orca %mrci soc DoSOC true # DoSSC true # MagneticField true # default false B 1,10,100,1000 # Strengh of the magnetic field in Gauss. # 4000 is the default value # Optional printing of the wave function for each # magnetic field settings MagneticField_PrintLevel 0 # default (disabled) TPrint 1e-3 end end ``` Then, the output contains three sets of data of splittings of the magnetic levels with the magnetic field applied parallel to x, y, and z directions: ``` End B (Gauss) Energy levels (cm-1) and populations for B || x 1.0 -0.030 0.333 0.012 0.333 0.018 0.333 10.0 -0.030 0.333 0.012 0.333 0.018 0.333 100.0 -0.031 0.333 0.012 0.333 0.020 0.333 1000.0 -0.102 0.333 0.012 0.333 0.091 0.333 B (Gauss) Energy levels (cm-1) and populations for B || y 1.0 -0.030 0.333 0.012 0.333 0.018 0.333 10.0 -0.030 0.333 0.012 0.333 0.018 0.333 100.0 -0.032 0.333 0.014 0.333 0.018 0.333 1000.0 -0.105 0.334 0.018 0.333 0.087 0.333 B (Gauss) Energy levels (cm-1) and populations for B || z 1.0 -0.030 0.333 0.012 0.333 0.018 0.333 10.0 -0.030 0.333 0.011 0.333 0.018 0.333 100.0 -0.030 0.333 0.005 0.333 0.025 0.333 1000.0 -0.079 0.333 -0.030 0.333 0.108 0.333 ``` Here the number in a row represents the strength of the magnetic field (in Gauss), and the following pairs of numbers denote the energy of the magnetic level (in cm$^{-1})$ with its occupation number. This table can be readily plotted with any suitable graphical program. (sec:mrci.soc.picturechange.detailed)= ### Relativistic Picture Change in Douglas-Kroll-Hess SOC and Zeeman Operators The DKH correction to the SOC operator is implemented in ORCA as a correction to the one-electron part of the SOMF operator. The DKH transformation is performed up to the second order, and the two-electron part in our implementation is left untransformed. However, the electronic density employed for evaluating the SOMF matrix elements is obtained from a scalar relativistic calculation. The inclusion of the DKH correction is controlled by the picturechange key in the rel block: ```orca %rel method DKH # relativistic method picturechange 2 # include the DKH correction to SOC end ``` The "picturechange" key can be set to 0, 1, and 2 for no picture change, the first order, and the second order DKH transformations of the SOC operator. With "picturechange" set to 1 or 2 the DKH correction are applied in the first order to the Zeeman operator. This correction has a visible effect on calculated g-tensors for molecules containing third-row and heavier atoms. (sec:mrci.soc.x-ray.spectroscopy.detailed)= ### X-ray Spectroscopy Likewise to the CASCI/NEVPT2 computational protocol presented in section {ref}`sec:casscf.CoreExitedStates.detailed` starting from ORCA 4.2 the MRCI module can be used to compute core excited spectra, namely X-ray absorption (XAS) and resonant inelastic scattering (RIXS) spectra. As discussed in the case of CASCI/NEVPT2 protocol {ref}`sec:casscf.CoreExitedStates.detailed` a similar strategy is followed to compute XAS/RIXS spectra within the MRCI module. In principle the XAS/RIXS spectra calculations require two steps: - In a first step one needs to optimize the valence active space orbitals in the framework of SA-CASSCF calculations, e.g. including valence excited states in the range between 6 to 15 eV. - In a second step the relevant core orbitals are rotated into the active space and the MRCI problem is solved by saturating the excitation space with singly core-excited electronic configurations using the previously optimized sets of orbitals. - The core orbitals are also included in the XASMOs definition. The use of this keyword is two fold. At first it effecteively reduces the number of the generated configuration state functions (CSFs) to those that exclusively contain contributions from the defined core orbitals. In the case of RIXS also XES (see below) the specified XASMOs are used to define intermediate or core ionized states. A representative input for the case of Fe(Cl) $_4$ is provided bellow: - In the first step one performs a SA-CASSCF calculation for the 5 and 15 quintet and triplet states (FeIICl4.casscf.inp). ```orca !CC-PWCVTZ-DK cc-pVTZ/C RIJCOSX SARC/J TightSCF DKH2 %rel FiniteNuc true end %basis newgto Cl "cc-pVTZ-DK" end newauxgto Cl "cc-pVTZ/C" end end %method FrozenCore FC_NONE end %casscf nel 6 norb 5 mult 5,3 nroots 5,15 switchstep nr end * xyz -2 5 Fe -17.84299991694815 -0.53096694321123 6.09104775508499 Cl -19.84288422845700 0.31089495619796 7.04101319789001 Cl -17.84298666758073 0.11868125024595 3.81067954087770 Cl -17.84301352218429 -2.87052442818457 6.45826391412877 Cl -15.84311566482982 0.31091516495189 7.04099559201853 * ``` - In a second step the core orbitals are rotated in the active space and the MRCI problem is solved by saturating the excitation space with all the quintet and triplet states that involve single excitations from the core orbitals (FeIICl4-mrci.inp) ```orca !MORead CC-PWCVTZ-DK cc-pVTZ/C RIJCOSX SARC/J TightSCF DKH2 %moinp "FeIICl4-casscf.gbw" %rel FiniteNuc true end %method FrozenCore FC_NONE end %scf rotate { 6,42,90} { 7,43,90} { 8,44,90} end end %basis newgto Cl "cc-pVTZ-DK" end newauxgto Cl "cc-pVTZ/C" end end %casscf nel 12 norb 8 mult 5,3 nroots 34,195 maxiter 1 switchstep nr end %mrci CIType MRCI intmode fulltrafo XASMOs 42, 43, 44 newblock 5 * nroots 34 excitations cisd refs CAS(12,8) end end newblock 3 * nroots 195 excitations cisd refs CAS(12,8) end end maxiter 100 soc printlevel 3 dosoc true end end * xyz -2 5 Fe -17.84299991694815 -0.53096694321123 6.09104775508499 Cl -19.84288422845700 0.31089495619796 7.04101319789001 Cl -17.84298666758073 0.11868125024595 3.81067954087770 Cl -17.84301352218429 -2.87052442818457 6.45826391412877 Cl -15.84311566482982 0.31091516495189 7.04099559201853 * ``` In a similar fashion Multireference Equation of Motopn Couple Cluster MR-EOM-CC (see next section) can also be used to compute X-ray spectra. Further information can be found in reference{cite}`2019xasmrcimreom` As it is explicitly described in the respective ROCIS section RIXS spectra can be requested by the following keywords: ```orca RIXS true # Request RIXS calculation (NoSOC) RIXSSOC true # Request RIXS calculation (with SOC) Elastic true # Request RIXS calculation (Elastic) ``` Please consult section {ref}`sec:rocis.rixs.detailed` for processing and analyzing the generated spectra Likewise to TDDFT ({ref}`sec:excitedstates.tddft.typical`) ROCIS ({ref}`sec:rocis.general.detailed`) and CASSCF ({ref}`sec:casscf.CoreExitedStates.detailed`) the computed transition densities also in the presence of SOC can be taken beyond the dipole approximation by using the [OPS tool]({ref}`sec:ops.detailed`) for details. 1. by performing a multiple expantion up to second order 2. by computing the exact transition moments The whole set of spectroscopy tables can be requested with the following commands: ```orca %mrci DoDipoleLength true DoDipoleVelocity true DoHigherMoments true DecomposeFoscLength true DecomposeFoscVelocity true DoFullSemiclassical true end ``` More details can be found in TDDFT ({ref}`sec:excitedstates.tddft.typical`) ROCIS ({ref}`sec:rocis.general.detailed`) and CASSCF ({ref}`sec:casscf.CoreExitedStates.detailed`) sections. Starting from ORCA 4.2 the previously reported RASCI-XES protocol reference{cite}`2014xesrasci`, which can compute K$_\beta$ Mainline XES spectra, can be processed entirely within the ORCA modules. In ORCA 5.0 a similar protocol (CASCI-XES) exist in the CASSCF module ({ref}`sec:casscf.CoreExitedStates.detailed`) - Like above or in the CASCI/NEVPT2 case in a first step one needs to optimize the valence active space orbitals in the framework of SA-CASSCF calculations, e.g. including valence excited states in the range between 6 to 15 eV for the N electron system. - In a second step the metal 1s and 3p orbitals are rotated in the active space and the 1s MO is defined in the XASMOs list - Computes the XES spectrum in the RASCI framework for the N-1 electron system in the presence of SOC if the XESSOC keyword for all the states that are dominated by 3p-1s electron decays. A representative input sequence for the case of Fe(Cl) $_6$ is provided bellow: As described in reference{cite}`2014xesrasci` at first for a CAS(5,5) the excitation space is saturated by the sextet as well as the 24 quartet and the 75 doublet states which are optimized in the SA-CASSCF fashion. ```orca !ZORA def2-TZVP def2-TZVP/C %cpcm epsilon 80 refrac 1.33 surfacetype gepol_ses end %scf MaxDisk 40000 end %casscf nel 5 norb 5 mult 6,4,2 nroots 1,24,75 shiftup 0.5 shiftdn 0.5 trafostep RI maxiter 150 end *xyzfile -3 6 Fe 0.0000 0.0000 0.000000 Cl 2.478 0.0000 0.000 Cl -2.478 0.0000 0.000 Cl 0.000005 2.478 0.00000 Cl 0.000005 -2.478 -0.0000 Cl -0.000 -0.000 2.478 Cl 0.000 -0.0000 -2.478 * ``` In following the 1s and 3p Fe based MOs are rotated in the active space and the XES spectra are computed for the \[Fe(Cl) $_6$\]$^+$ system for the 4 septet and 81 quintet states. ```orca ! ZORA def2-TZVP def2-TZVP/C noiter moread AllowRHF %moinp "fecl6_casscf.gbw" %cpcm epsilon 80 refrac 1.33 surfacetype gepol_ses end %scf MaxDisk 40000 end %scf rotate {0,59,90} {36, 60, 90} {37,61,90} {38,62,90} end end %mrci citype mrci UseIVOs false Etol 1e-5 newblock 5 * excitations none nroots 81 refs ras(12:4 1/5/ 0 0) end end newblock 7 * excitations none nroots 4 refs ras(12:4 1/5/ 0 0) end end XASMOs 59 soc dosoc true XESSOC true end end *xyzfile -2 7 Fe 0.0000 0.0000 0.000000 Cl 2.478 0.0000 0.000 Cl -2.478 0.0000 0.000 Cl 0.000005 2.478 0.00000 Cl 0.000005 -2.478 -0.0000 Cl -0.000 -0.000 2.478 Cl 0.000 -0.0000 -2.478 * ``` As a result the X-ray emission spectrum is calculated and the intensities are computed on the basis of the transition electric dipole moments ``` Printing the XES spectrum ... ------------------------------------------------------------------------------------- SPIN-ORBIT X-RAY EMISSION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS ------------------------------------------------------------------------------------- Transition Energy INT TX TY TZ 1 421 -> 5 7228.632 0.000000000000 0.00000 0.00000 0.00000 2 422 -> 5 7228.632 0.000000000000 0.00000 0.00000 0.00000 3 423 -> 5 7228.632 0.000000000000 0.00000 0.00000 0.00000 4 424 -> 5 7228.632 0.000000000000 0.00000 0.00000 0.00000 5 425 -> 5 7228.632 0.000000000000 0.00000 0.00000 0.00000 ... 242 422 -> 25 7177.286 0.000917305388 0.00025 0.00171 0.00149 243 423 -> 25 7177.286 0.002043577370 0.00197 0.00211 0.00181 244 424 -> 25 7177.286 0.000789769987 0.00114 0.00133 0.00119 245 425 -> 25 7177.286 0.000026130790 0.00018 0.00034 0.00002 246 426 -> 25 7177.286 0.000035191741 0.00034 0.00028 0.00003 247 427 -> 25 7177.286 0.005143175830 0.00294 0.00345 0.00296 248 428 -> 25 7177.341 0.000000000000 0.00000 0.00000 0.00000 249 429 -> 25 7177.341 0.000000000001 0.00000 0.00000 0.00000 250 430 -> 25 7177.341 0.000000000001 0.00000 0.00000 0.00000 251 431 -> 25 7177.341 0.000000000000 0.00000 0.00000 0.00000 252 432 -> 25 7177.341 0.000000000000 0.00000 0.00000 0.00000 ... 4991 431 -> 420 7153.111 0.000195885011 0.00106 0.00000 0.00000 4992 432 -> 420 7153.111 0.002719228427 0.00256 0.00299 0.00002 All Done ------------------------------------------------------------------------------------- ``` The resulted XES spectrum can be visualized by processing the above output file with the `orca_mapspc` ```orca orca_mapspc fecl6_xes.out XESSOC -x07140 -x17190 -w4.0 -eV -n10000 ``` This will result in {numref}`fig:RASCI-XESSOC`. (fig:RASCI-XESSOC)= ```{figure} ../../images/RASCI_XESSOC.* Calculated RASCI K$_\beta$ XES spectrum of [Fe(Cl) $_6$]$^{+}$ . ```