(sec:rocis.detailed)= # Excited States via ROCIS and DFT/ROCIS The ORCA program package includes the `orca_rocis` module to perform configuration interaction with single excitations (CIS) calculations using a restricted open-shell Hartee-Fock (ROHF) reference function. It produces excitation energies, absorption energies and CD intensities. It was designed with the aim to reproduce and - even more importantly - reliably predict transition metal L-edges as observed in X-ray absorption spectroscopy (XAS). (sec:rocis.general.detailed)= ## General Use In the present implementation the `orca_rocis` module is only able to perform CIS calculations on top of a high-spin ROHF reference function. All spins of the unpaired electrons have to be coupled ferrmoagnetically to give a total spin of $S = \frac{1}{2}N$, where $N$ is the number of unpaired electrons. Other ROHF functions such as Zerner's configuration averaged or spin averaged ROHF cannot be used as reference. The input for a high spin ROHF calculation is done in the `%scf` block. ```orca %scf HFTyp ROHF # Flag for ROHF ROHF_Case HighSpin # selects the high-spin case ROHF_NEl[1] = 4 # the number of unpaired electrons end ``` In our experience ROHF calculations suffer a lot from convergence problems. UHF calculations generally exhibit better convergence properties. In most cases the quasi-restricted orbitals (qro's) of a UHF calculation resemble the ROHF orbitals. Thus the program features the ability to start a ROCIS calculation on top of a UHF calculation. It will automatically create the qro's and build the reference determinant with them. If one wants to avoid the (small) errors that are introduced by this procedure, one may take the qro's of a UHF calculation as starting orbitals for a subsequent ROHF calculation. Furthermore it is possible to invoke the `orca_rocis` module for closed-shell molecules. The program will then perform a CI calculation with the provided RHF reference function. In this case it will yield the same result as the `orca_cis` program. A number of basic variables in the `%rocis` block control the settings of the Davidson procedure that is used to solve the CI problem: ```orca %rocis NRoots 6 # number of desired roots MaxDim 5 # Davidson expansion space = MaxDim * NRoots ETol 1e-6 # energy convergence tolerance RTol 1e-6 # residual vector convergence tolerance MaxIter 35 # maxmimum number of iterations NGuessMat 512 # dimension of the guess matrix: 512x512 end ``` The dimension of the iterative subspace is given by MaxDim $\cdot$ NRoots. The lowest possible choice for MaxDim is a value of 2. In general, by choosing `MaxDim` $\approx$ 5-10 times `NRoots` you will achieve a more favorable convergence by the cost of an increased disk space requirement. Increasing the `NGuessMat` variable will improve the convergence of the iterative CI procedure. The amount of output produced during the calculation is controlled via the `PrintLevel` variable ```orca %rocis NRoots 3 PrintLevel 3 end ``` Note, that this does not influence which spectra are calculated or printed. The absorption spectrum calculated on the basis of the pure dipole approximation for your calculation is always printed. In addition, it is possible to allow for electric quadrupole and magnetic dipole contributions to the absorption spectrum as well as to calculate the CD spectrum check section ({ref}`sec:ops.detailed`) for details. By defining in the %rocis block: ```orca %rocis NRoots 6 DoDipoleLength true DoDipoleVelocity true DoHigherMoments true DecomposeFoscLength true DecomposeFoscVelocity true DoFullSemiclassical true DoCD true end ``` The printed spectra look like this: ``` ----------------------------------------------------------------------------- ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS ----------------------------------------------------------------------------- State Energy Wavelength fosc T2 TX TY TZ (cm-1) (nm) (au**2) (au) (au) (au) ----------------------------------------------------------------------------- 1 2635.0 3795.1 0.000000001 0.00000 0.00001 -0.00001 0.00029 2 4365.5 2290.7 0.000011416 0.00086 0.01200 -0.00864 0.02534 3 4368.2 2289.3 0.000011174 0.00084 -0.02006 0.01442 0.01523 4 5977.9 1672.8 0.000093897 0.00517 -0.04164 -0.05863 0.00000 5 65245.3 153.3 0.027669631 0.13961 -0.20555 -0.31203 -0.00023 ----------------------------------------------------------------------------- ABSORPTION SPECTRUM VIA TRANSITION VELOCITY DIPOLE MOMENTS ----------------------------------------------------------------------------- State Energy Wavelength fosc P2 PX PY PZ (cm-1) (nm) (au**2) (au) (au) (au) ----------------------------------------------------------------------------- 1 2635.0 3795.1 0.000000085 0.00000 -0.00000 0.00000 -0.00004 2 4365.5 2290.7 0.001777771 0.00005 -0.00315 0.00223 -0.00618 3 4368.2 2289.3 0.001850956 0.00006 0.00526 -0.00372 -0.00371 4 5977.9 1672.8 0.003237195 0.00013 0.00667 0.00937 0.00000 5 65245.3 153.3 0.057301314 0.02555 0.08779 0.13358 0.00010 ------------------------------------------------------------------- CD SPECTRUM ------------------------------------------------------------------- State Energy Wavelength R MX MY MZ (cm-1) (nm) (1e40*sgs) (au) (au) (au) ------------------------------------------------------------------- 1 2635.0 3795.1 0.00007 -0.00511 -0.01539 0.00021 2 4365.5 2290.7 10.02484 0.57434 -0.40490 0.42899 3 4368.2 2289.3 -10.03730 0.34432 -0.24269 -0.71470 4 5977.9 1672.8 0.01537 -0.00033 -0.00032 -0.00286 5 65245.3 153.3 -0.00865 0.00004 0.00003 -0.00005 ----------------------------------------------------------------------------------------------------- COMBINED ELECTRIC DIPOLE + MAGNETIC DIPOLE + ELECTRIC QUADRUPOLE SPECTRUM ----------------------------------------------------------------------------------------------------- State Energy Wavelength D2 m2 Q2 D2+m2+Q2 D2/TOT m2/TOT Q2/TOT (cm-1) (nm) (*1e6) (*1e6) ----------------------------------------------------------------------------------------------------- 1 2635.0 3795.1 0.00000 0.00011 0.00000 0.00000000080469 0.86010 0.13938 0.00052 2 4365.5 2290.7 0.00001 0.47866 0.00000 0.00001189497194 0.95976 0.04024 0.00000 3 4368.2 2289.3 0.00001 0.48629 0.00000 0.00001166062671 0.95830 0.04170 0.00000 4 5977.9 1672.8 0.00009 0.00001 0.00001 0.00009389664707 1.00000 0.00000 0.00000 5 65245.3 153.3 0.02767 0.00000 0.06183 0.02766969236508 1.00000 0.00000 0.00000 ----------------------------------------------------------------------------------------------------- COMBINED ELECTRIC DIPOLE + MAGNETIC DIPOLE + ELECTRIC QUADRUPOLE SPECTRUM (origin adjusted) ----------------------------------------------------------------------------------------------------- State Energy Wavelength D2 m2 Q2 D2+m2+Q2 D2/TOT M2/TOT Q2/TOT (cm-1) (nm) (*1e6) (*1e6) ----------------------------------------------------------------------------------------------------- 1 2635.0 3795.1 0.00000 0.00000 0.00000 0.00000000069409 0.99716 0.00016 0.00268 2 4365.5 2290.7 0.00001 0.38277 0.00039 0.00001179947536 0.96753 0.03244 0.00003 3 4368.2 2289.3 0.00001 0.36798 0.00045 0.00001154275975 0.96808 0.03188 0.00004 4 5977.9 1672.8 0.00009 0.00000 0.00001 0.00009389663928 1.00000 0.00000 0.00000 5 65245.3 153.3 0.02767 0.00003 0.06176 0.02766969232228 1.00000 0.00000 0.00000 ``` Furthermore like in TD-DFT (section {ref}`sec:excitedstates.tddft.typical`) or CASSCF one may obtain intensities by evaluating the 2nd order oscillation strengths, or the full semi-classical oscillation strengths. - The exact oscillation strengths behave like the multipole expansion in the velocity representation. - They are by definition origin independent they do not suffer from artificial negative values like the multipole moments beyond 1st order. - They are used with the multipole moments up to 2nd order to regenerate the electric dipole, electric quadrupole and magnetic dipole contributions in either length or the velocity representation. For the Fe K-edge XAS spectrum of \[FeCl$_4$\]$^{2-}$. This will result in addition to the following tables for the velocity representation: ``` ------------------------------------------------------------------------------------------------------------- COMBINED ELECTRIC DIPOLE + MAGNETIC DIPOLE + ELECTRIC QUADRUPOLE SPECTRUM (Origin Independent, Velocity) ------------------------------------------------------------------------------------------------------------- State Energy Wavelength P2 m2 Q2 P2+m2+Q2+PM+PO P2/TOT m2/TOT Q2/TOT (cm-1) (nm) (*1e6) (*1e6) ------------------------------------------------------------------------------------------------------------- 1 57131638.5 0.2 0.00000 0.00000 3.75184 0.00000375184371 0.00000 0.00000 1.00000 2 57131638.5 0.2 0.00000 0.00000 3.75184 0.00000375184267 0.00000 0.00000 1.00000 3 57145543.6 0.2 0.00007 0.00000 3.46619 0.00007086820341 0.95853 0.00000 0.04891 4 57145543.6 0.2 0.00007 0.00000 3.46620 0.00007078008474 0.95972 0.00000 0.04897 5 57145543.6 0.2 0.00007 0.00000 3.46620 0.00007084079919 0.95889 0.00000 0.04893 11 57351031.6 0.2 0.00000 0.00000 0.00000 0.00000000000002 0.99463 0.00618 0.00216 12 57351031.6 0.2 0.00000 0.00000 0.00000 0.00000000000001 0.00000 0.00000 0.00000 13 57351031.6 0.2 0.00000 0.00000 0.00000 0.00000000000002 0.99414 0.00692 0.00217 15 57354687.7 0.2 0.00000 0.00000 0.00000 0.00000000000888 0.00898 0.00000 0.00002 ------------------------------------------------------------------------------------------------------------- COMBINED ELECTRIC DIPOLE + MAGNETIC DIPOLE + ELECTRIC QUADRUPOLE SPECTRUM (Exact Formulation, Velocity) ------------------------------------------------------------------------------------------------------------- State Energy Wavelength P2 m2 Q2 Exact Osc. Strength P2/TOT m2/TOT Q2/TOT (cm-1) (nm) (*1e6) (*1e6) ------------------------------------------------------------------------------------------------------------- 1 57131638.5 0.2 0.00000 0.00000 3.02719 0.00000302719471 0.00000 0.00000 1.00000 2 57131638.5 0.2 0.00000 0.00000 2.66225 0.00000266224706 0.00000 0.00000 1.00000 3 57145543.6 0.2 0.00007 0.00000 3.46619 0.00007092969904 0.95853 0.00000 0.04891 4 57145543.6 0.2 0.00007 0.00000 3.46620 0.00007074406444 0.95972 0.00000 0.04897 5 57145543.6 0.2 0.00007 0.00000 3.46620 0.00007075200792 0.95889 0.00000 0.04893 11 57351031.6 0.2 0.00000 0.00000 0.00000 0.00000000000002 0.99463 0.00618 0.00216 12 57351031.6 0.2 0.00000 0.00000 0.00000 0.00000000000001 0.98256 0.01631 0.00209 13 57351031.6 0.2 0.00000 0.00000 0.00000 0.00000000000002 0.99414 0.00692 0.00217 15 57354687.7 0.2 0.00000 0.00000 0.00000 0.00000000001200 0.00898 0.00000 0.00002 .... ``` These spectra are plotted by calling: ```orca orca_mapspc MyOutput.out ABS/ABSV/CD/ABSQ/ABSOI/ABSVOI -eV -x0(start) -x1(stop) -w(width) -n(points) ``` In particular ABSOI and ABSVOI will plot the exact transition moments spectra at the Length and Velocity representations (For the multiple expansion contributions). If calculations on large molecules are conducted, the integral transformation will be the most time-consuming part. Therefore it is strongly recommended to use the resolution of the identity (RI) approximation in those cases. It effectively reduces the computational costs of the transformation step by only introducing minor errors to the calculation. It has to be kept in mind that in order to keep the introduced errors small, one has to provide a reasonable auxiliary basis sets along with your normal basis set input. Starting from ORCA 4.0 the basis set definition on ORCA has changed. This also affects the definition of the auxiliary basis set when the DoRI keyword is set. ROCIS will then only allow in the mainline `/C` auxiliary basis sets to be set (i.e. def2-TZVP`/C`). As these basis are usually optimised on the presence of effective core potentials (ECPs) they are generally not recommended for core-electron calculations. The `/J` auxiliary basis set need to be used and they are specified in the following way. ```orca %basis AuxC "def2/J" end ``` ```{literalinclude} ../../orca_working_input/C06S17_455.inp :language: orca ``` The `orca_rocis` module provides two ways of choosing the orbital excitation space: by orbital energy or orbital number. In the former case an energy window has to be specified and the program will then take all orbitals, whose orbital energies lie within this window, into account. Note, that one actually has to define two orbital windows: One for the donor and the second for the acceptor orbital. The input of the windows is done as an array: The first two numbers define the donor space while the last two numbers define the acceptor space. ```orca %rocis NRoots 3 EWin = -5,5,-5,5 end ``` The default is to keep core orbitals and very high lying virtual orbitals out of their respective orbital excitation spaces. Since these orbitals span a space that is usually not reachable with regular UV/Vis spectroscopy, this is a reasonable approximation. One has to keep in mind that an orbital energy window makes only sense if the orbitals used in the calculation have a well-defined orbital energy. As a consequence one cannot use an orbital energy window for a calculation with localized orbitals. The second way to specify the excitation space is by orbital numbering. ```orca %rocis NRoots 3 OrbWin = 1,13,9,22 end ``` In restricted calculations only one set of spatial orbitals is created. Hence it is not necessary to provide orbital windows for $\alpha$ and $\beta$ electrons separately. Of course, only doubly or singly occupied orbitals can act as donor orbitals and only singly and nonoccupied orbitals can act as acceptor orbitals. The program recognises nonoccupied orbitals in the donor space and doubly occupied orbitals in the acceptor space and removes both. The many-electron expansion space of a ROCIS calculation in ORCA is divided into five classes. Using second quantised replacement operators $E_{p}^{q} =\hat{{a} }_{q\alpha }^{\uparrow } \hat{{a} }_{p\alpha } +\hat{{a} }_{q\beta }^{\uparrow } \hat{{a} }_{p\beta }$ they take the form{cite}`roemelt2013`. $$\begin{array}{l} \left| \Phi_i^s \right\rangle= E_{i}^{s} \left| 0 \right\rangle\\ \left| \Phi_s^a \right\rangle= E_{s}^{a} \left| 0 \right\rangle\\ \left| \Phi_i^a \right\rangle= \frac{1}{\sqrt 2 }E_{i}^{a} \left| 0 \right\rangle\\ \left| \Phi_{ti}^{as} \right\rangle=E_{t}^{a} E_{i}^{s} \left| 0 \right\rangle\\ \left| \Phi_{ti}^{as} \right\rangle= \frac{1}{\sqrt 6 }\left( E_{i}^{a} -2E_{s}^{a} E_{i}^{s} \right)\left| 0 \right\rangle\\ \end{array} $$ (eqn:217) The orbital label $i$ denotes a doubly occupied orbital, $s$ and $t$ refer to singly occupied orbitals and orbital label $a$ corresponds to a virtual orbital. The form of the excitation classes ensures that all excited states are eigenfunctions of the $\hat{{S} }^{2}$-operator and have the same total spin $S$ as the electronic ground state. Each of the five excitation classes can be switched on or off manually. ```orca %rocis NRoots 3 Do_is true # Include DOMO->SOMO excitations Do_sa true # Include SOMO->Virtual excitations Do_ia true # Include DOMO->Virtual excitations Do_ista true # Include DOMO->SOMO coupled to # SOMO->Virtual excitations with s not equal t Do_isa true # Include DOMO->SOMO coupled to # SOMO->Virtual excitations with s = t # --------------------------------- # by default all switches for the # excitation classes are set to # ``true'' # --------------------------------- end ``` Formally, the $\left| \Phi_{ti}^{as} \right\rangle$ and $\left| \Phi_{ti}^{at} \right\rangle$ excitation classes can be regarded as double excitations. When the program finishes the ROCIS calculation it gives the excitation energy together with the composition for each root. According to the number of labels of the respective functions $\left|\Phi \right\rangle$, contributions from excited configuration state functions belonging to the different excitation classes are given by two, three or four numbers. ```orca STATE 5 Exc. Energy: 297.279mEh 8.089eV 65245.3cm**-1 47->50 : 0.2196 47->51 : 0.0138 37->50 : 0.1165 41->50 : 0.0960 38->46 ; 47->50 : 0.0103 37->46 ->50 : 0.0150 37->47 ->50 : 0.0938 37->48 ->50 : 0.0179 37->49 ->50 : 0.0179 41->46 ->50 : 0.0174 41->47 ->50 : 0.0585 41->48 ->50 : 0.0213 41->49 ->50 : 0.0211 ``` Furthermore the `orca_rocis` module is able to calculate the effect of spin-orbit coupling (SOC) on the calculated ground and excited states. It introduces SOC in the framework of quasi-degenerate perturbation theory (QDPT). The SOC Hamiltonian is diagonalized in the basis of the calculated ROCIS states $\left| \Psi_I^{SM} \right\rangle$, where $I$ is the root label and $S$ and $M$ are the spin and magnetic spin quantum numbers, respectively{cite}`neese1998inorgchem`, {cite}`roemelt2013`. ```orca %rocis NRoots 3 OrbWin = 1, 3 ,9 ,22 SOC true # invokes the calculation of SOC effects TEMPERATURE 10 # temperature for SOC corrected spectra in Kelvin end ``` After the SOC calculation the program will produce additional spectra for the SOC corrected results. The spectra contain transitions from the $2S+1$ lowest lying states into all excited states, where S is the spin quantum number of the electronic ground state. These $2S+1$ lowest states may be split up in the order of 1-100 cm$^{-1}$. Due to the small magnitude of the splitting, all of the $2S+1$ states can be significantly populated even at low temperatures. Experimentally, the intensity of a given transition is dependent on the population of the corresponding initial state. With the `TEMPERATURE` keyword the population of the theoretically calculated states can be manipulated by the varying the fictive temperature of the system. It has to be mentioned that the electric quadrupole transitions between spin-orbit coupled states are not well defined and are likely to give unreasonable results. Hence it is recommended to use the `DoHigherMoments` keyword only for calculations that do not include SOC. ``` ------------------------------------------------------------------------------- SPIN ORBIT CORRECTED ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS ------------------------------------------------------------------------------- States Energy Wavelength fosc T2 TX TY TZ (cm-1) (nm) (au**2) (au) (au) (au) ------------------------------------------------------------------------------- 0 1 5.6 0.0 0.000000000 0.00000 0.00003 0.00002 0.00000 0 2 6.2 0.0 0.000000000 0.00000 0.00000 0.00000 0.00005 0 3 23.7 422287.3 0.000000000 0.00000 0.00000 0.00000 0.00000 0 4 23.7 421562.8 0.000000000 0.00000 0.00018 0.00025 0.00000 0 5 2621.7 3814.3 0.000000000 0.00000 0.00000 0.00001 0.00005 0 6 2622.0 3813.9 0.000000000 0.00000 0.00003 0.00012 0.00000 0 7 2634.7 3795.5 0.000000095 0.00002 0.00388 0.00273 0.00049 0 8 2634.9 3795.2 0.000000103 0.00002 0.00039 0.00027 0.00495 0 9 2639.5 3788.6 0.000000001 0.00000 0.00001 0.00001 0.00036 0 10 4223.6 2367.6 0.000000103 0.00002 0.00043 0.00029 0.00390 0 11 4223.9 2367.5 0.000000120 0.00002 0.00348 0.00236 0.00046 0 12 4296.3 2327.6 0.000000696 0.00010 0.00562 0.00842 0.00000 0 13 4357.6 2294.8 0.000000002 0.00000 0.00001 0.00001 0.00049 0 14 4418.1 2263.4 0.000005778 0.00083 0.00653 0.00468 0.02762 0 15 4422.1 2261.4 0.000005517 0.00079 0.02184 0.01559 0.00832 0 16 4488.2 2228.0 0.000000001 0.00000 0.00004 0.00006 0.00038 0 17 4524.2 2210.3 0.000000001 0.00000 0.00030 0.00018 0.00000 0 18 4597.2 2175.2 0.000000027 0.00000 0.00023 0.00016 0.00191 0 19 4597.4 2175.2 0.000000051 0.00001 0.00213 0.00153 0.00023 0 20 6043.6 1654.6 0.000047989 0.00502 0.04104 0.05779 0.00000 0 21 6049.5 1653.0 0.000000014 0.00000 0.00109 0.00057 0.00001 0 22 6051.3 1652.5 0.000000021 0.00000 0.00001 0.00004 0.00150 0 23 6069.7 1647.5 0.000000000 0.00000 0.00005 0.00007 0.00000 0 24 6069.9 1647.5 0.000000028 0.00000 0.00098 0.00138 0.00000 0 25 65281.7 153.2 0.014223474 0.13787 0.20423 0.31010 0.00023 0 26 65281.7 153.2 0.000000035 0.00000 0.00032 0.00048 0.00011 0 27 65281.7 153.2 0.000009000 0.00009 0.00522 0.00774 0.00001 0 28 65281.7 153.2 0.000007207 0.00007 0.00460 0.00698 0.00000 0 29 65281.7 153.2 0.000047448 0.00046 0.01179 0.01791 0.00001 1 2 0.6 0.0 0.000000000 0.00000 0.00001 0.00001 0.00000 1 3 18.1 553477.5 0.000000000 0.00000 0.00000 0.00000 0.00009 1 4 18.1 552233.6 0.000000000 0.00000 0.00006 0.00004 0.00000 1 5 2616.1 3822.5 0.000000063 0.00001 0.00006 0.00003 0.00261 1 6 2616.4 3822.1 0.000000060 0.00001 0.00211 0.00144 0.00006 1 7 2629.1 3803.6 0.000000143 0.00002 0.00225 0.00321 0.00003 1 8 2629.3 3803.3 0.000000002 0.00000 0.00015 0.00025 0.00040 1 9 2633.9 3796.7 0.000000271 0.00003 0.00011 0.00008 0.00538 1 10 4218.0 2370.8 0.000000005 0.00000 0.00031 0.00046 0.00019 ... ``` If the `PrintLevel` value is set to 3 or higher, the program will print out the composition of the SOC corrected states in the basis of states $\left| \Psi_I^{SM} \right\rangle$. ```orca Eigenvectors of SOC calculation: the threshold for printing is: 0.010000 weight : Root Spin Ms State 0: 0.00 cm**-1 0.00000 eV 0.378045 : 0 2 2 0.235825 : 0 2 0 0.378045 : 0 2 -2 State 1: 5.61 cm**-1 0.00070 eV 0.496236 : 0 2 2 0.496236 : 0 2 -2 State 2: 6.20 cm**-1 0.00077 eV 0.496291 : 0 2 1 0.496291 : 0 2 -1 ``` Further details of the SOC calculation such as the procedure of SOC integral calculation can be controlled via the `%rel` block (section {ref}`sec:relativistic.detailed`. (sec:rocis.transMetal.detailed)= ## Transition Metal L-Edges with ROCIS or DFT/ROCIS The `orca_rocis` program was designed to calculate transition metal L-edge spectra of large molecules as they are observed in X-ray absorption spectroscopy (XAS). An L-edge results when an electron is promoted from the 2p shell of a transition metal ion into the valence d shell by an X-ray photon. Strong spin-orbit coupling in the 2p shell and p-d coupling phenomena complicate the interpretation and even more so the prediction of these spectra. It has to be kept in mind that the present program applies a variety of approximations which might lead to observable deviations from experimentally determined spectra. However, we believe that the results obtained from the program are in general qualitatively correct and in most cases accurate close to the experimental uncertainty. In cases where quantitative accuracy is not met, the provided results might still give some insight into the mechanisms of intensity distribution in the spectra. The special input structure for orbital windows described in {ref}`sec:rocis.general.detailed` allows the user to restrict the donor orbital space to the transition metal 2p shell. The acceptor orbital space is the same as in regular UV/Vis spectroscopy. It should include all singly occupied molecular orbitals and as many virtual orbitals as one can afford in the calculation. The number of roots should be chosen large enough so that at least all 2p-3d single excitations are calculated. In many cases even more roots are required since doubly excited or charge transfer states may become important. Moreover the strong SOC apparent in the 2p shell of transition metal ions necessitates the additional calculation of excited states with a total spin of $S' = S + 1$ and $S' = S -1$ where $S$ is the total spin of the electronic ground state. Accordingly four additional excitation classes introduce excited configuration state functions with a lower and higher spin multiplicity. They feature the second quantized spin raising and lowering operators $\hat{{S} }_{pq}^{+} =\hat{{a} }_{q\alpha}^{\uparrow} \hat{{a} }_{p\beta }$, $\hat{{S} }_{pq}^{-} =\hat{{a} }_{q\beta}^{\uparrow } \hat{{a} }_{p\alpha }$. $$ \begin{array}{r} \left. \begin{array}{l} \left| \Phi_{i}^{\left(t-\right)} \right\rangle =\sqrt{ \frac{2{S}'+1}{2{S}'+2} } S_{ti}^{-} \left| 0 \right\rangle-\sum\limits_{u \neq t}^{\text{SOMO}} {\frac{1}{\sqrt{2{S}'+1}}\frac{1}{\sqrt{2{S}'+2} }S_{uu}^{-} E_{i}^{t} \left|0 \right\rangle} \\ \left| \Phi_{i}^{\left(t-\right)} \right\rangle =\sqrt{ \frac{2{S}'+1}{2{S}'+2} } S_{ti}^{-} \left| 0 \right\rangle-\sum\limits_{u \ne t}^{\text{SOMO} } { \frac{1}{\sqrt{ 2{S}'+1} }\frac{1}{\sqrt{ 2{S}'+2} }S_{uu}^{-} E_{i}^{t} \left|0 \right\rangle} \\ \left| \Phi_{i}^{\left(a-\right)} \right\rangle =\sqrt{ \frac{2{S}'+1}{2{S}'+3} } S_{ai}^{-} \left| 0 \right\rangle-\sum\limits_{t}^{\text{SOMO} } { \sqrt{\frac{\left({ S}' +1\right)^2 - { S}'^2}{\left({ S}'+1\right)\left(2{S}'+3\right)} }\frac{1}{\sqrt{ 2\left(2{S}'+2\right)} } S_{tt}^{-} E_{i}^{a} \left|0 \right\rangle}\\ \hspace{2cm} + \sum\limits_{t, u \ne t}^{\text{SOMO} } { \sqrt{\frac{2}{\left(2{S}'+2\right)\left(2{S}'+3\right)} }\sqrt{\frac{1}{\left(2{S}'+2\right)2\left(2{S}'+1\right)} } S_{tt}^{-} S_{uu}^{-} S_{ai}^{+} \left|0 \right\rangle} \end{array} \right\} \quad S' = S-1 \\ \left. \left| \Phi_{i}^{a^+} \right \rangle = S_{ai}^{+} \left| 0 \right \rangle \right\} \quad { S}' = S +1 \end{array} $$ (eqn:218) Inclusion of configuration state functions with higher or lower multiplicity is invoked with the keywords `DoLowerMult` and `DoHigherMult`, respectively. ```orca %rocis NRoots 20 SOC true DoRI true PrintLevel 3 DoLowerMult true #Invokes a CI calculation #with S'=S-1 DoHigherMult true #Invokes a CI calculation #with S'=S+1 OrbWin = 6,8,0,2000 end ``` The program will conduct a separate Davidson procedure for each multiplicity. Subsequently it gives the excitation energies and compositions of the calculated excited states for all included multiplicities. After all CI calculations are finished, the program gives a list of all calculated roots with their excitation energies and their multiplicities. It is this number that will be referred to as label $I$ in the decomposition of spin-orbit coupled states in the basis $\left| \Psi_{I}^{SM} \right\rangle$. It is very important to note, that when states with different multiplicities are calculated this number might deviate from the number that appears in the respective CI part of the output. If one gets confused about the numbering of the states, the state energies might act as a guideline through the output of the program. Without SOC the spin exclusion rule applies which means that only excited states with a total spin equal to the ground state spin ($S' = S$) give rise to non-vanishing intensities. Hence, only these transitions are listed in the spectra before SOC. ``` -------------------------------------------------------------------------------- ROOT Mult Excitation energy[Eh] [cm-1] [eV] -------------------------------------------------------------------------------- 0 5 0.00000000 0.00 0.000 1 5 26.24822856 5760820.28 714.251 2 5 26.24833619 5760843.90 714.254 3 5 26.27159871 5765949.43 714.887 4 5 26.27982129 5767754.08 715.110 5 5 26.30321870 5772889.22 715.747 6 5 26.30458669 5773189.46 715.784 7 5 26.33143414 5779081.79 716.515 8 5 26.33600432 5780084.83 716.639 9 5 26.33865219 5780665.97 716.711 10 5 26.34522494 5782108.52 716.890 11 5 26.34577552 5782229.36 716.905 12 5 26.35183534 5783559.34 717.070 13 3 26.42121780 5798787.03 718.958 14 3 26.42122881 5798789.45 718.958 ... 42 7 27.22926558 5976133.02 740.946 43 7 27.23201078 5976735.52 741.021 44 7 27.23280499 5976909.83 741.042 45 7 27.23594814 5977599.67 741.128 46 7 27.23865050 5978192.77 741.201 47 7 27.26590445 5984174.32 741.943 48 7 27.26597947 5984190.78 741.945 49 7 27.26604364 5984204.87 741.947 50 3 27.29447169 5990444.10 742.720 51 3 27.30121861 5991924.88 742.904 52 3 27.30655497 5993096.08 743.049 53 3 27.30685328 5993161.55 743.057 54 3 27.31274496 5994454.62 743.218 55 7 27.52164817 6040303.58 748.902 56 7 27.52433114 6040892.42 748.975 57 7 27.52448641 6040926.50 748.979 58 7 27.53903479 6044119.50 749.375 59 7 27.53935644 6044190.10 749.384 ------------------------ ROCIS-EXCITATION SPECTRA ------------------------ NOTE: At this point no SOC is included!!! Hence only transitions to states with the same spin multiplicity as the ground state are observed!!! Center of mass = ( -0.0011, -0.0021, 0.0000) Calculating the Dipole integrals ... done Transforming integrals ... done Calculating the Linear Momentum integrals ... done Transforming integrals ... done ----------------------------------------------------------------------------- ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS ----------------------------------------------------------------------------- State Energy Wavelength fosc T2 TX TY TZ (cm-1) (nm) (au**2) (au) (au) (au) ----------------------------------------------------------------------------- 1 5760820.3 1.7 0.000985130 0.00006 0.00612 -0.00434 0.00011 2 5760843.9 1.7 0.000777158 0.00004 -0.00008 0.00006 0.00666 3 5765949.4 1.7 0.000000036 0.00000 0.00000 0.00001 -0.00004 4 5767754.1 1.7 0.000007564 0.00000 0.00033 0.00057 -0.00000 5 5772889.2 1.7 0.025379335 0.00145 -0.00031 0.00021 -0.03804 6 5773189.5 1.7 0.026898175 0.00153 0.03203 -0.02254 -0.00039 7 5779081.8 1.7 0.000000323 0.00000 -0.00006 -0.00009 -0.00008 8 5780084.8 1.7 0.001711738 0.00010 -0.00572 -0.00805 0.00001 9 5780666.0 1.7 0.113054940 0.00644 -0.04616 -0.06564 -0.00001 10 5782108.5 1.7 0.151287595 0.00861 0.00073 -0.00052 0.09281 11 5782229.4 1.7 0.147199895 0.00838 0.07488 -0.05266 -0.00088 12 5783559.3 1.7 0.000000026 0.00000 0.00001 -0.00001 0.00004 28 5960986.7 1.7 0.004292708 0.00024 -0.00881 -0.01263 -0.00000 29 5963084.1 1.7 0.001638281 0.00009 -0.00774 0.00553 0.00006 30 5963136.7 1.7 0.001369356 0.00008 -0.00005 0.00003 -0.00869 31 5963484.9 1.7 0.000935993 0.00005 0.00415 0.00587 -0.00000 32 5968477.0 1.7 0.000661255 0.00004 0.00493 -0.00349 -0.00007 33 5968705.6 1.7 0.000607238 0.00003 0.00006 -0.00004 0.00579 35 5970943.7 1.7 0.000000001 0.00000 0.00000 0.00000 -0.00001 ``` After calculation of SOC in the basis of all calculated ROCIS roots, the program prints out the composition of the spin-orbit coupled states (if `PrintLevel` \>2) and the corresponding absorption spectrum. ``` Eigenvectors of SOC calculation: the threshold for printing is: 0.010000 weight : Root Spin Ms State 0: 0.00 cm**-1 0.00000 eV 0.129027 : 0 2 2 0.741116 : 0 2 0 0.129027 : 0 2 -2 ``` ``` ------------------------------------------------------------------------------- SPIN ORBIT CORRECTED ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS ------------------------------------------------------------------------------- States Energy Wavelength fosc T2 TX TY TZ (cm-1) (nm) (au**2) (au) (au) (au) ------------------------------------------------------------------------------- 0 1 0.0 0.0 0.000000000 0.00000 0.00000 0.00000 0.00000 0 2 0.8 0.0 0.000000000 0.00000 0.00000 0.00000 0.00000 0 3 0.8 0.0 0.000000000 0.00000 0.00000 0.00000 0.00000 0 4 1.0 0.0 0.000000000 0.00000 0.00000 0.00000 0.00000 0 5 5729330.4 1.7 0.000080556 0.00002 0.00013 0.00009 0.00464 0 6 5729330.4 1.7 0.000096984 0.00003 0.00415 0.00295 0.00013 0 7 5731365.3 1.7 0.000000001 0.00000 0.00001 0.00000 0.00000 0 8 5731365.4 1.7 0.000000000 0.00000 0.00000 0.00000 0.00001 0 9 5733452.5 1.7 0.000058329 0.00002 0.00323 0.00227 0.00004 0 10 5733477.2 1.7 0.000066389 0.00002 0.00003 0.00002 0.00421 0 11 5734964.4 1.7 0.000000034 0.00000 0.00005 0.00007 0.00004 0 12 5737151.2 1.7 0.000047769 0.00001 0.00208 0.00291 0.00000 ``` With the aid of the `orca_mapspc` program it is possible to extract a `.plt` file from the printed spectra, which then can be used to generate a plot of the intensity vs the excitation energy. The `orca_mapspc` program applies Gaussian type lineshape functions to the calculated transitions with a user-defined FWHM. One has to provide some information for the program such as the name of the output file, the type of spectrum you wish to plot, the energy range and the like. It is invoked in the command line and the parameters are given as arguments: ```orca orca_mapspc FeIICl4.out socabs -eV -w1 -n3000 -x0710 -x1740 ``` The first argument has to be the output file of your calculation followed by the type of spectrum that should be plotted. In the case of transition metal L-edges it is an absorption spectrum after the SOC correction. The arguments "-eV" (use electron Volt as energy unit), "-w1" (FWHM $=$ 1eV), "-n3000" (use 3000 grid points), "-x0710" and "-x1740" (energy range: 710 to 740 eV) have to be adapted to the specific calculation. As a result, one obtains a `.plt` and a `.stk` file. The `.plt `file contains five columns. In the first column one finds the energy and in the second the total intensity. Columns three to five contain the x-,y- and z-components of the transition moment. Note, that the distribution of the transition moment among its spatial components depends on the orientation of your molecular axis system. The `.stk` file contains a list of all transitions with their respective transition energy and intensity. A more detailed description of the `orca_mapspc` program and its usage can be found in chapter {ref}`sec:utilities.mapspc.detailed`. (fig:711)= ```{figure} ../../images/711.* Comparison of the experimentally observed (black) and calculated ROCIS (red) Fe L-edge of [FeCl$_4$]$^{2-}$. The red bars highlight the contribution of individual states to the total spectrum. The calculation was performed using the TZVP basis set. ``` For many transition metal compounds the description of the electronic ground and excited states by Hartree-Fock theory and CIS is of rather poor quality. Especially covalency and relative spin state energetics are not reproduced correctly. This in turn might lead to wrong intensity distributions in the calculated L-edge spectra. In the majority of these cases the quality of the description and hence the predicted L-edge spectra can be significantly improved with the DFT/ROCIS method{cite}`roemelt2013`. It features the usage of a restricted open-shell Kohn-Sham matrix as reference and also uses the DFT orbitals for setting up the excited configuration state functions in the CI expansion. The two electron integrals that include the DFT orbitals are scaled according to their nature and their position in the CI matrix by the parameters $c_{1}$, $c_{2}$ and $c_{3}$. They all lie in the interval \[0;1\]. Parameters $c_{1}$ and $c_{2}$ scale coulomb- and exchange- like terms in the diagonal part of the CI matrix, whereas $c_{3}$ reduces the size of all off-diagonal elements of the CI matrix. For example: $$\begin{array}{l} H_{ia,ia}^{\text{DFT/ROCIS} } =F_{aa}^{C\left( \text{KS} \right)} -F_{ii}^{C\left( \text{KS} \right)} -c_{1} \left({ ii\vert aa} \right)+2c_{2} \left({ ia\vert ia} \right) \\ H_{ia,jb}^{\text{DFT/ROCIS} } =c_{3} \left\{{ \delta_{ij} F_{ab}^{C\left( \text{KS} \right)} -\delta_{ab} F_{ji}^{C\left( \text{KS} \right)} -\left({ ij\vert ab} \right)+2\left({ ia\vert jb} \right)} \right\} \end{array} $$ (eqn:219) The three default parameters $c_{1} = 0.18$, $c_{2} = 0.20$ and $c_{3} = 0.40$ have been optimized for a test set of molecules and their excited states on a B3LYP/def2-TZVP(-f) level of theory but can be freely chosen{cite}`roemelt2013`. It is most likely that for a different combination of test molecules, functional and basis set, a different set of parameters gives better results. Since the parameters are chosen with regard of a good \"balance\" between orbital energies, Coulomb and exchange integrals, a new set of parameters should at least crudely resemble their relative proportions. ```orca ! B3LYP def2-TZVP(-f) TightSCF %Basis AuxC "def2/J" end %ROCIS NRoots 20 DoRI true SOC true DoHigherMult true PrintLevel 3 OrbWin = 5,7,50,60 DoDFTCIS true #switches on the DFT/ROCIS method DFTCIS_c = 0.18, 0.20, 0.40 #Array input of the three parameters end ``` (fig:712)= ```{figure} ../../images/712.* Comparison of the experimentally observed (black) and calculated (red) Ti L-edge of [Cp$_2$TiCl$_2$]. The red bars highlight the contribution of the individual states to the total spectrum. The pure ROCIS method (left) predicts a wrong L$_3$-L$_2$ intensity ratio and strongly overestimates the splitting of the satellite features to the main bands. Better results are obtained with the DFT/ROCIS method (right). ``` (sec:rocis.natTransOrb.detailed)= ## Natural Transition Orbitals/ Natural Difference Orbitals Likewise to CIS and TD-DFT (section {ref}`sec:tddft.natTransOrb.detailed`) The nature of the calculated excited states in ROCIS and DFT/ROCIS can be analyzed by using the Natural Transition Orbitals (NTO) or Natural Difference Orbitals (NDO) machineries.{cite}`felix2014` Note that: - The NTO analysis is based on the transition density between ground and excited states. Hence is valid for singly excited states and for states of the same multiplicity. - The NDO analysis on the otherhand is somewhat more flexible in this respect as it is based on the difference density between ground and excited states. - Presently, only one analysis (NTO or NDO) can be performed at a time while when both flags are on the NTO analysis switches off. An example is given below for \[FeCl$_4$\]$^{2-}$: ```orca !B3LYP def2-TZVP Conv TightSCF LargePrint PAL4 %Basis AuxC "def2/J" end %ROCIS NRoots 40 PrintLevel 3 MaxCore 4000 MaxDim 360 SOC true DoRI true DoNTO true DoNDO true NDOThresh/NTOThresh 1e-4 NDOStates/NTOStates= 1,2,3,4,5,6,7,8,9,10,13,14,15 DoLowerMult true DoHigherMult true DoDFTCIS true DFTCIS_c = 0.18, 0.20, 0.40 OrbWin = 6,8,0,2000 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 * ``` Then the respective NTO and NDO analysis for state 15 is given below: ``` ------------------------------------------ NATURAL TRANSITION ORBITALS FOR STATE 14 ------------------------------------------ done Solving eigenvalue problem for the Occupied space ... done Solving eigenvalue problem for the Acceptor space ... done Natural Transition Orbitals were saved in nto.14.nto Threshold for printing occupation numbers 1.0e-04 E= 25.447756 au 692.469 eV 5585137.0 cm**-1 49[0] -> 46[1] : n= 0.39056909 48[0] -> 47[1] : n= 0.08619374 47[0] -> 48[1] : n= 0.00441125 ``` ``` ------------------------------------------------- NATURAL DIFFERENCE ORBITALS FOR STATE 14 ----------------------------------------------- done Solving eigenvalue problem for the Occupied space ... done Solving eigenvalue problem for the Acceptor space ... done Natural Difference Orbitals were saved in ndo.14.ndo Threshold for printing occupation numbers 1.0e-04 E= 25.447756 au 692.469 eV 5585137.0 cm**-1 49[0] -> 46[1] : n= 0.81173217 48[0] -> 47[1] : n= 0.17903699 47[0] -> 48[1] : n= 0.01165859 46[0] -> 49[1] : n= 0.00922738 45[0] -> 50[1] : n= 0.00112567 ``` For closed shell cases the orbitals are save in similar way to TDDFT and CIS (section {ref}`sec:tddft.natTransOrb.detailed`). In the case of open shell cases for convenience donor orbitals are saved with orbital operator 0 while acceptor orbitals with orbital operator 1. This needs to be specified in the `orca_plot` program and should not be confused with the `spin-up` and `spin-down` orbitals in the UHF and UKS cases. In practice one can use this machinery to analyze for example the relativistically corrected states located at 705.5 eV (when shifted with respect to experiment). It can be seen that these states contain for example significant contributions from state 14. NTO or NDO analysis then shows that this state is dominated by the spin conserving DOMO-SOMO $2p_z-3d_{yz}$ single electron excitation. (fig:ROCIS_NTO)= ```{figure} ../../images/ROCIS_NTO.* DFT/ROCIS calculated L3 XAS spectrum of [Fe(Cl) $_4$]$^{2-}$ together with NDO analysis for state 14. Constant broadening $0.5$ eV and isovalue for the orbital plots $0.03$ a.u. is used throughout ``` (sec:rocis.rixs.detailed)= ## Resonant Inelastic Scattering Spectroscopy (sec:rocis.rixs.general.detailed)= ### General Starting from ORCA version 4.0 ROCIS module can be used to calculate RIXS spectra The present implementation is directly based on the Kramers Heisenerg Dirac (KDH) expression formula for near resonant and resonant conditions $$\left|{ {\alpha _{\rho \lambda } }({E_{ex} },{E_{sc} }) } \right|_{Total}^2 = { \sum\limits_F { \left|{ \sum\limits_V { \frac{{\left\langle F \right|{ m_\rho }\left| V \right\rangle\left\langle V \right|{ m_\lambda }\left| I \right\rangle} }{{{E_{VI} } - { E_{ex} } - i\frac{1}{2}{\Gamma _V} }} } } \right|} ^2}\left\{{ \frac{{{\Gamma _F} }}{{{{({E_{FV} } - { E_{ex} } + { E_{sc} }) }^2} + \frac{1}{4}{\Gamma _F}^2} }} \right\}$$ $$\left|{ {\alpha _{\rho \lambda } }({E_{ex} },{E_{sc} },V) } \right|_{resonant}^2 = \sum\limits_F { {{\left|{ \left\langle F \right|{ m_\rho }\left| V \right\rangle} \right|}^2}{{\left|{ \left\langle V \right|{ m_\lambda }\left| I \right\rangle} \right|}^2} } f({E_{VI} },{E_{FV} },{E_{ex} },{E_{sc} },{\Gamma _V},{\Gamma _F})$$ $$\left|{ {\alpha _{\rho \lambda } }({E_{VI} },{E_{sc} }) } \right|_{Direct}^2 = \sum\limits_V { \left|{ {\alpha _{\rho \lambda } }({E_{VI} },{E_{sc} },V) } \right|_{resonant}^2}$$ The resonance scattering cross section for total and direct cases, averaged over all orientations of the molecule and integrated over all directions and polarizations of scattered radiation is given in equations: $$\sigma _{_{RXES} }^{Total}({E_{ex} },{E_{sc} }) = \frac{{8\pi E_{sc}^3{E_{ex} }} }{{9{c^4} }}\sum\limits_{\rho ,\lambda = x,y,z} { \left|{ {\alpha _{\rho \lambda } }({E_{ex} },{E_{sc} }) } \right|_{Total}^2}$$ $$\sigma _{_{RXES} }^{Direct}({E_{ex} },{E_{sc} }) = \frac{{8\pi E_{sc}^3{E_{ex} }} }{{9{c^4} }}\sum\limits_{\rho ,\lambda = x,y,z} { \left|{ {\alpha _{\rho \lambda } }({E_{ex} },{E_{sc} }) } \right|_{Direct}^2}$$ Interference effects can be then derived in a straightforward way from equation: $$\sigma _{RXES}^{interference}(E_{ex}^{},E_{sc}^{}) = \sigma _{RXES}^{Total}(E_{ex}^{},E_{sc}^{}) - \sigma _{RXES}^{Direct}(E_{ex}^{},E_{sc}^{})$$ In order to access RIXS spectroscopy in the ROCIS module one needs in addition to specify a 2nd donor space. This is specified by defining an OrbWin array with 6 elements: The first four elements define the ranges of the two donor spaces while the last two elements the respective acceptor space range. ```orca OrbWin = 0,0,2,4,45,60 ``` An important difference with respect to the conventional ROCIS or DFT/ROCIS calculations is the fact that two donor spaces of very different energy ranges are involved (e.g. K-edge, L-edge) which requires to restrict somewhat the acceptor space and saturate it with as many states as possible. The main calling commands in order to perform a RIXS calculation within both ROCIS and CASSCF blocks are the following: - RIXS true. Similar to absorption spectroscopy, this requests the RIXS calculation to be performed based on the calculated non-relativistic ground state multiplicity States - RIXSSOC true. By turning-on this flag the RIXS is calculated by taking in account the relativistically corrected Ms States. - Elastic true. This flag indicates whether the resonant condition in which the initial and Final states coincide should be taken into account. Note that the intensity of this spectral feature might be overestimated as presently the non resonant terms are not treated The respective ROCIS input reads then as follows: ```orca !B3LYP def2-TZVP SlowConv %Basis AuxC "def2/J" end %ROCIS NRoots 200 PrintLevel 3 MaxCore 4000 DoRI true DoHigherMult true SOC true RIXS true # Request RIXS calculation (NoSOC) RIXSSOC true # Request RIXS calculation (with SOC) Elastic true # Request RIXS calculation (Elastic) DoDFTCIS true DFTCIS_c =0.18,0.20,0.40 OrbWin = 2,4,25,33,0,100 end * xyzfile 2 2 test.xyz ``` When running the calculation one can monitor if the requested NRoots were sufficient enough to select the states dominated by both the donor orbital spaces ``` -------------------------------------------------------------------------------- ROOT Mult Excitation energy[Eh] [cm-1] [eV] -------------------------------------------------------------------------------- 0 2 0.00000000 0.00 0.000 1 2 0.06611737 14511.08 1.799 2 2 0.07728471 16962.03 2.103 3 2 0.07732428 16970.72 2.104 ... 84 2 33.75471831 7408304.35 918.513 85 2 33.77073325 7411819.22 918.948 86 2 33.77076955 7411827.19 918.949 87 4 34.06882971 7477243.83 927.060 88 2 34.07021441 7477547.74 927.098 ... ``` If that is not the case the respective RIXS calculations will not be performed and a Warning Message will be generated: ```orca Making the RIXS files ... WARNING!: Flag for RIXS property calculation was identified but there is zero number of Intermediate and/or Final states: No Cross-Section properties will be evaluated ...Skipping this part TIP: Increase the number of NRoots and/or decrease or increase the acceptor orbital space ...Done ``` A successful run on the other hand will generate the following messages for RIXS and RIXSSOC calculations. ``` ---------------------------------------------------------------------------------- ROCIS RIXS SPECTRUM ---------------------------------------------------------------------------------- Making the RIXS data files for Inelastic and Elastic Scattering Ground State: 1 Intermediate States: 21 Final States: 59 The RIXS cross section will be generated from: 60 Ground-Final State Pairs and 21 Intermediate States/Pair Calculating Intensities... 10% done 20% done 30% done 40% done 50% done 60% done 70% done 80% done 90% done 100% done Storing the files...All Done ---------------------------------------------------------------------------------- ``` ``` ---------------------------------------------------------------------------------- ROCIS RIXSSOC SPECTRUM ---------------------------------------------------------------------------------- Making the RIXS-SOC data files for Inelastic and Elastic Scattering Ms States: 2 Intermediate States: 78 Final States: 214 The RIXS cross section will be generated from: 432 Ground-Final State Pairs and 78 Intermediate States/Pair Calculating Intensities... 10% done 20% done 30% done 40% done 50% done 60% done 70% done 80% done 90% done 100% done Storing the files...All Done ---------------------------------------------------------------------------------- ``` In both cases the number of involved Initial, Final and Intermediate states is specified explicitly. For example in the case of RIXSSOC 2 Ms Ground states, 78 Intermediate states and 214 Final states are involved. Then the RIXS cross section for elastic and inelastic scattering will be generated by 432 (2\*(2+214)) Ground-Final State-Pairs and 78 Intermediate States per Ground-Final state pair. (sec:rocis.rixs.mapspc.detailed)= ### Processing the spectra with `orca_mapspc` By calling `orca_mapspc` with the following keywords: ```orca orca_mapspc test.el_inel.rocis.rixssoc RIXS -x0871 -x1876 -x2-1 -x34 -w0.4 -g0.4 -l -n125 -m125 -dx20 -eaxis1 ``` The program will process the `test.el_inel.rocis.rixssoc` file with the following parameters: Energy range along x : 871-876 eV Energy range along y: -1-34 eV -l indicates Lorentzian broadening Width along x (gamma): 0.4 eV Width along y (gamma): 0.4 eV Points along x: 125 Points along y:125 Shift to be applied along Incident energy/Emission axis: 20 eV The y axis will be Energy Transfer axis. If -eaxis2 is the y axis will be then Emission Energy axis All this information is printed during the data processing: ``` Mode is RIXS Using Lorentzian shape Cannot read the paras.inp file ... taking the line width parameter from the command line Cannot read the udex.inp file ... taking the excitation energy ranges from the command line Cannot read the udem.inp file ... taking the emission energy ranges from the command line Cannot read the gfsp.inp file ... No Ground-Final State Pairs will be evaluated --------------------------------------------------------------------------------- PLOTTING RIXS SPECTRA --------------------------------------------------------------------------------- Input File : test.el_inel.rocis.rixssoc Incident Energy Excitation axis : 871.000 ... 876.000 eV 125 points Energy transfer axis : -1.000 ... 4.000 eV 125 points Incident Energy Shift : 20.000 eV Lorenzian Linewidth along Incident Axis : 0.400 eV Lorenzian Linewidth along Energy Transfer/Emission Axis : 0.400 eV y axis : 1 -> Energy transfer Number of user defined cuts at constant Excitation Energy axis: 0 Number of user defined cuts at constant Emission/Energy Transfer Energy axis : 0 Making checks...Done Proccessing data... 10% done 20% done ... 100% done RIXS-plotting done Incident Energy range: 845.800 ... 869.249 Emission/Energy-transfer range: 0.000 ... 4.853 Now storing the 2D file... Done Making the Integrated spectra along Energy Transfer/Emission axis... Done Making the Integrated spectra along Incident axis... Done All Done --------------------------------------------------------------------------------- ``` Successful run will generate the following files: The RIXS planes of the Total, Direct and Interference RIXS intensity as indicated in the above equations: ``` test.el_inel.rocis.rixssoc.total_rixs.dat test.el_inel.rocis.rixssoc.direct_rixs.dat test.el_inel.rocis.rixssoc.interference_rixs.dat ``` In addition one obtains the integrated spectra at constant Incident energies (CIE): ``` test.el_inel.rocis.rixssoc.dw.dat ``` as well as at constant Emission/Energy Transfer energies (CEE/CET): ``` test.el_inel.rocis.rixssoc.wex.dat ``` (fig:ROCIS_RIXS)= ```{figure} ../../images/ROCIS_RIXS.* DFT/ROCIS calculated RIXS planes for ${[Cu{(N{H_3})_4}]^{2 - } }$. Left: Total RIXS Intensity, Middle: Direct RIXS intensity and Right: Interference RIXS intensity. Lorentzian lineshape broadening with constant widths along Incident and Energy Transfer axis (0.5 and 0.2 eV respectively) were used throughout. ``` (sec:rocis.rixs.cut.detailed)= ### Generating Cuts Cuts along x and y axis can be generated with two ways: 1\) At first, this action can be performed by adding the following keywords: `uex` and `udw` accounting for generating cuts at constant Incident Energies (CIE) and at constant Emission (CEE)/or at constant Energy Transfer (CET) respectively, together with the desired number of cuts. 2\) Alternatively, the energies of the desired cuts can be specified as lists in the files udex.inp (user defined excitations) udem.inp (user defined emissions) For example if in udex.inp one specifies: ```orca 872.5 874.2 ``` and for the cuts along Energy Transfer axis one just specify -udw3 ```orca orca_mapspc test.el_inel.rocis.rixssoc RIXS -x0871 -x1876 -x2-1 -x34 -w0.4 -g0.4 -l -n125 -m125 -dx20 -eaxis1 -udw3 ``` Then at the end one gets: ``` Making the specified cuts (2) at constant Excitation Energy axis... Writing file: test.el_inel.rocis.rixssoc_872.50.rxes_vs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_872.50.rxes_fs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_874.20.rxes_vs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_874.20.rxes_fs.dat ...Done Done Making the specified cuts (3) at constant Emission/Energy Transfer axis... Writing file: test.el_inel.rocis.rixssoc_-1.00.xas_vs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_-1.00.xas_fs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_1.50.xas_vs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_1.50.xas_fs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_4.00.xas_vs.dat ...Done Writing file: test.el_inel.rocis.rixssoc_4.00.xas_fs.dat ...Done Done All Done --------------------------------------------------------------------------------- ``` The files `*_rxes_fs.dat` are RXES spectra containing all individual contributions from all Final states together with the Direct, the Total and the Interference contributions at the given constant Incident Energy. Similarly, the `*_rxes_vs.dat` are RXES spectra containing individual contributions of the Intermediate states, together with the Direct the Total and the Interference contributions at the given constant Incident Energy Likewise, the respective `*_xas_fs.dat` and `*_xas_vs.dat` are XAS type spectra with individual contributions at a given constant Emission or Energy transfer Energy These files are Energy vs Intensity files and read like: 1\) for `*fs.dat` ``` X S- 1( 0- 0) S- 2( 0- 1) DIRECT TOT INTERFERENCE ``` 2\) for `*vs.dat` ``` X S- 1( 45) S- 2( 47) DIRECT TOT INTERFERENCE ``` In the first case S -1(0-0) represents the individual contribution of a given Ground-Final state pair. The numbering follows the numbering of the output file e.g.: ``` Eigenvalues: cm-1 eV Boltzmann populations at T = 300.000 K 0: 0.0000 0.0000 3.44e-01 1: 8.3818 0.0010 3.31e-01 ``` Hence, in this case S -1 represents the elastic scattering intensity. In the second case S -1(45) represents the individual contribution of a given Intermediate state. ``` 44: 66918.6071 8.2968 1.43e-140 45: 6996678.8061 867.4775 0.00e+00 46: 6996693.0276 867.4793 0.00e+00 ``` In this case S -1 represents the intensity contribution of the first Intermediate state. Starting from ORCA 4.2 in every RIXS requested calculation the Off resonant XES spectrum is automatically generated in every RIXS requested calculation. ``` ---------------------------------------------------------------------------------- ROCIS RIXS SPECTRUM ---------------------------------------------------------------------------------- Making the RIXS data files for Inelastic and Elastic Scattering Ground State: 1 Intermediate States: 28 Final States: 588 The RIXS cross section will be generated from: 589 Ground-Final State Pairs and 28 Intermediate States/Pair The Off-Resonance XES spectrum will be printed Calculating Intensities... 10% done 20% done 30% done 40% done 50% done 60% done 70% done 80% done 90% done 100% done Printing the XES spectrum and Storing the files... ------------------------------------------------------------------------------------- X-RAY EMISSION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS ------------------------------------------------------------------------------------- Transition Energy INT TX TY TZ (eV) (fosc) (au) (au) (au) ------------------------------------------------------------------------------------- 1 589 -> 0 6403.377 0.000000000721 0.00000 0.00000 0.00000 2 590 -> 0 6403.380 0.000000000083 -0.00000 0.00000 0.00000 3 591 -> 0 6403.685 0.000873238810 0.00236 0.00000 0.00000 4 592 -> 0 6404.766 0.000000000154 0.00000 0.00000 0.00000 5 593 -> 0 6408.288 0.000000006850 -0.00001 0.00000 0.00000 6 594 -> 0 6408.295 0.000034710300 -0.00047 0.00000 0.00000 ... 16490 614 -> 588 6387.989 0.000000000000 0.00000 0.00000 0.00000 16491 615 -> 588 6388.222 0.000000000000 0.00000 0.00000 0.00000 16492 616 -> 588 6388.881 0.000000000000 0.00000 0.00000 0.00000 All Done ---------------------------------------------------------------------------------- ``` Hence also the myfile-rixs.out file can also be processed with the `orca_mapspc` to generate the respective XES spectra: ```orca orca_mapspc myfile_rixs.out XES/XESSOC -x06000 -x16500 -w2.0 -eV -n10000 ``` (sec:rocis.PNO.detailed)= ## Core PNO-ROCIS, PNO-ROCIS/DFT It has been shown recently{cite}`2018pnorocis` that it is possible to combine the powerful machinery of the PNOs with the ROCIS and ROCIS/DFT methods to formulate the core PNO-ROCIS and PNO-ROCIS/DFT methods. The usage of PNOs here is somewhat unconventional since they are not used to treat electron correlation effects in a state specific manner. Rather, the PNOs are used to identify the relevant part of the virtual space that can be reached by excitation out of local core orbitals. This subspace of the virtual space is local, thus leading to a linear scaling, state universal method. The PNO-ROCIS calculations can be requested with the following keywords: ```orca ... DoPNO true #Flag to call the PNO truncation TCutPNO 1e-11#Threshold to cutout the PNO populations XASElems 0 #Number of the involved element to the calculated core XAS calculation OrbWin = 0,0,0,2000 ... ``` As has been shown in reference{cite}`2018pnorocis` a universal TCutPNO 1e-11 threshhold can be defined for all edges provided that the PNOs are constructed by taking into account all the availiable core orbitals in the systems. For example in the case of a 1st row transition metal this will be the 9 1s, 2s, 2p, 3s and 3p MOs. These orbitals will be identified automatically by the program provided that the element or the elements for which the XAS calculation will be performed are specified within the XASElems keyword. In the following example these correspond to Core MOs 36-44. Note that the CoreMOs list should not be confused with the OrbWin which is used to specify the excitation space that will be actually used in the actual calculation. ``` =============================================== Core PNO/ROCIS truncation ================================================ ------------------------------------------------ Calculating Integrals... ------------------------------------------------ ... ------------------------------------------------ Calculating Guess Amplitudes and Densities... ------------------------------------------------ ---------------------------------------------------------------- The densities will be generated from the Detected Core MOs: ---------------------------------------------------------------- MO= 36, E= -261.246087 Eh MO= 37, E= -31.777896 Eh MO= 38, E= -27.263122 Eh MO= 39, E= -27.263122 Eh MO= 40, E= -27.263122 Eh MO= 41, E= -3.914132 Eh MO= 42, E= -2.457405 Eh MO= 43, E= -2.457405 Eh MO= 44, E= -2.457405 Eh ``` Alternativelly one can also use the CoreMOs keyword to individual select the respective CoreMOs ```orca ... DoPNO true #Flag to call the PNO truncation TCutPNO 1e-11#Threshold to cutout the PNO populations CoreMOs 0,1,6,7,8,29,30,31,32 #The core MOs for the selected element #to perform the XAS calculation OrbWin = 0,0,0,2000 ... ``` A complete list of CoreMOs of the different atoms can be found in reference{cite}`2018pnorocis` The program will then proceed and generate the Core PNOs and use the TCutPNO threshold to reduce the Virtual MO space. In the following example only virtual orbitals are selected out of the total 1445 virtual MOs ``` TCutPNO: 1.000e-11 Virtual orbitals before selection: 368 ... 1812 (1445 MO's) Virtual orbitals after selection: 368 ... 447 ( 80 MO's) PNO transformation completed in: 177.09 sec ``` From this point and on the programm will proceed the usual way. This will result in extraordinary computation speeding ups without loss in accuracy. (sec:rocis.magProp.detailed)= ## ROCIS Magnetic Properties Several magnetic properies are availiable in the ROCIS method Including g-tensors (G-Matrix), zero field splittings (ZFS), hyperfine couplings (HFCs) and electric field gradients (EFGs). The g-tensors as well as the zfs are calculated on the basis of the Effective Hamiltonian as well in the sum over states (SOS) framework. HFCs are calculated in the SOS framework while EFGs are calculated as expectation values. Please consult also the respective discussion in the MRCI chapter (section {ref}`sec:mrci.detailed`) ```orca ... DoHeff true # Requests calculation of G-tenosrs and ZFS # in the effective Hamiltonian framework DoEPR true # Requests calculation of G-tenosrs, ZFS and HFCs # in the Sum over states (SOS) framework AtensorNuc 0 # Nuclei to account for the HFCs calculation NAtensors 1 # How many Nuclei are included in the HFCs calculation ATensor 0 # Nucleus to calculate HFCs and EFGs NDoubGtensor 1 # Kramers doublets to account for the g tensor calculations ... ``` This will enter the calculation in the ROCIS Spin Hamiltonian section ``` -------------------------------------------------------- ROCIS SPIN HAMILTONIAN PROPERTIES -------------------------------------------------------- ``` (sec:rocis.keywords.detailed)= ## Keyword List ```orca %rocis #----------------------------------------------------------- # GENERAL KEYWORDS #----------------------------------------------------------- NRoots 3 # The number of desired roots MaxDim 5 # Davidson expansion space = MaxDim * NRoots MaxIter 35 # Maximum CI Iterations NGuessMat 512 # The dimension of the guess matrix ETol 1e-6 # Energy convergence tolerance RTol 1e-6 # Residual Convergence tolerance MaxCore 2000 # Maximum memory used during the calculation in MB EWin= -5,5,-5,5 # Energy Window that defines orbital excitation space OrbWin=6,8,0,2000 # Orbital Window that defines orbital excitation space # (overrides EWin) DoRI false # Switch for the RI approximation DoLoc false # Switch for localization of Donor orbital space LocMet PipekMezey # chooses the localization method: # PipekMezey or FosterBoys. # Abbreviations "PM" and "FB" # are equivalent to full names. SOC false # Switch for inclusion of SOC TEMPERATURE 10 # The fictive temperature for the # SOC corrected spectra DoDFTCIS false # Switch for the DFT/ROCIS method DFTCIS_C = 0.18, 0.20, 0.40 #Array Input of the # three DFT/ROCIS parameters #----------------------------------------------------------- # FLAGS FOR EXCITATION SPACES #----------------------------------------------------------- Do_is true # Include DOMO->SOMO excitations Do_sa true # Include SOMO->Virtual excitation Do_ia true # Include DOMO->Virtual excitations Do_ista true # Include DOMO->SOMO excitations # coupled to SOMO->Virtual # excitations with s not equal t Do_isa true # Include DOMO->SOMO excitations # coupled to SOMO->Virtual # excitations with s = t DoLowerMult false # Switch for excitation with S’=S-1 Do_LM_is true # Include DOMO->SOMO excitations # with S’=S-1 Do_LM_sa true # Include SOMO->Virtual excitations # with S’=S-1 Do_LM_ia true # Include DOMO->Virtual excitations # with S’=S-1 Do_LM_ss true # Include SOMO->SOMO excitations # with S’=S-1 DoHigherMult false # Switch for DOMO->Virtual # excitations with S’=S+1 #----------------------------------------------------------- OUTPUT KEYWORDS #----------------------------------------------------------- PrintLevel 3 # Controls the amount of output # produced during the calculation RIXS false # Perform a RIXS calculation RIXSSOC false # Perform a RIXS calculation on the basis # of relativistically corrected states Elastic false # Include the elastic line in the generation # of the RIXS or RIXSSOC spectra PlotDiffDens = 1,2 # Array input for plotting # difference densities of CI roots # 1 and 2 to the ground state. PlotSOCDiffDens=1,2 # Array input for plotting # difference densities of SOC # states 1 and 2 to the ground state DoNTO false # Request Natural Transition Orbital Analysis DoNDO false # Request Natural Difference Orbital Analysis # (if true it switches off the NTO analysis) NDOThresh 1e-4 # Threshold for printing occupation numbers NTOThresh 1e-4 # Threshold for printing occupation numbers NDOStates = 1,2 # Array input for states to be taken into account NTOStates = 1,2 # Array input for states to be taken into account TPrint 0.01 # Threshold for contributions to CI # and SOC states to be printed DoPNO false # Performs the calculation in the PNO-ROCIS framework DoCD true # Request circular dichroism calculation DoDipoleLength true # Request the use of electric moments in a length formulation DoDipoleVelocity true # Request the use of electric moments in a velocity formulation DoHigherMoments true # Request the calculation of electric quadrupole and magnetic # dipole moments contributions DoFullSemiclassical true # Request the calculation of complete semiclassical # multipolar moments DecomposeFoscLength true # Request the decomposition of the oscillator strengths # in a multipolar expansion under a length formulation DecomposeFoscVelocity true # Request the decomposition of the oscillator strengths # in a multipolar expansion under a velocity formulation ```