```{index} Electrical Properties, Electric Moments, Polarizabilities ``` (sec:spectroscopyproperties.elprop)= # Electrical Properties - Electric Moments and Polarizabilities A few basic electric properties can be calculated in ORCA although this has never been a focal point of development. The properties can be accessed straightforwardly through the `%elprop` block: ```{literalinclude} ../../orca_working_input/C05S13_222.inp :language: orca ``` The polarizability (dipole-dipole, dipole-quadrupole, quadrupole-quadrupole) is calculated analytically through solution of the coupled-perturbed (CP-)SCF equations for HF and DFT runs (see {ref}`sec:essentialelements.cpscf`) and through the CP-CASSCF equations for CASSCF runs (see {ref}`sec:spectroscopyproperties.casscfresp`). Analytic polarizabilities are also available for CCSD/CCSD(T) (see {ref}`sec:modelchemistries.autoci.autociresp`), RI-MP2 and DLPNO-MP2, as well as double-hybrid DFT methods. For canonical MP2 one can use AUTOCI for analytic calculations (see {ref}`sec:modelchemistries.autoci.autociresp`) or differentiate the analytical dipole moment calculated with relaxed densities. For other correlation methods only a fully numeric approach is possible. ```orca --------------------------------------------------- STATIC POLARIZABILITY TENSOR (Dipole/Dipole) --------------------------------------------------- Method : SCF Type of density : Electron Density Type of derivative : Electric Field (Direction=X) Multiplicity : 1 Irrep : 0 Relativity type : Basis : AO The raw cartesian tensor (atomic units): 12.852429555 -0.002199911 0.000000170 -0.002199911 12.860507003 -0.000000346 0.000000170 -0.000000346 12.868107945 diagonalized tensor: 12.851869269 12.861067290 12.868107945 Orientation: 0.969064588 -0.246807263 0.000017958 0.246807263 0.969064586 -0.000050696 -0.000004890 0.000053560 0.999999999 Isotropic polarizability : 12.86035 ``` As expected the polarizability tensor is isotropic. Dipole-quadrupole polarizability tensors are printed as a list of 18 different components, with the first index running over x,y,z and the second index running over xx,yy,zz,xy,xz,yz. This is known as the "pure Cartesian" version of the tensor, although other conventions may exist in the literature that differ from the ORCA values by a constant factor. ```orca --------------------------------------------------- STATIC POLARIZABILITY TENSOR (Dipole/Quadrupole) --------------------------------------------------- Method : SCF Type of density : Electron Density Type of derivative : Electric Field (Direction=X) Multiplicity : 1 Irrep : 0 Relativity type : Basis : AO The raw cartesian tensor (atomic units): X- X X : 11.577165985 X- Y Y : -5.795339382 X- Z Z : -5.797320742 X- X Y : 0.001285565 X- X Z : 0.000000155 X- Y Z : -0.000000077 Y- X X : 0.001386387 Y- Y Y : 8.200445841 Y- Z Z : -8.198375727 Y- X Y : -5.794687548 Y- X Z : 0.000000228 Y- Y Z : -0.000000121 Z- X X : -0.000000151 Z- Y Y : 0.000000627 Z- Z Z : -0.000000812 Z- X Y : -0.000000312 Z- X Z : -5.798359323 Z- Y Z : -8.205110537 ``` After this, the "traceless" version of the tensor is printed, which is usually denoted by $A_{x,xx}, A_{x,xy}$ etc. in the literature{cite}`McLean1967JCP,Pedersen2011JCC,Yan2020CTP`. This is the preferred format for reporting dipole-quadrupole polarizability tensors. Certain references use the opposite sign convention than reported here, but generally the conventions of traceless polarizability tensors are more unified than those of pure Cartesian polarizability tensors. ```orca ------------------------------------------------------------- STATIC TRACELESS POLARIZABILITY TENSOR (Dipole/Quadrupole) ------------------------------------------------------------- Method : SCF Type of density : Electron Density Type of derivative : Electric Field (Direction=X) Multiplicity : 1 Irrep : 0 Relativity type : Basis : AO The raw cartesian tensor (atomic units): X- X X : 17.373496046 X- Y Y : -8.685262003 X- Z Z : -8.688234043 X- X Y : 0.001928347 X- X Z : 0.000000232 X- Y Z : -0.000000116 Y- X X : 0.000351329 Y- Y Y : 12.298940512 Y- Z Z : -12.299291841 Y- X Y : -8.692031322 Y- X Z : 0.000000342 Y- Y Z : -0.000000181 Z- X X : -0.000000058 Z- Y Y : 0.000001109 Z- Z Z : -0.000001050 Z- X Y : -0.000000468 Z- X Z : -8.697538984 Z- Y Z : -12.307665806 ``` The quadrupole-quadrupole polarizability tensor is similarly printed in both the pure Cartesian and traceless forms. Again, the traceless form (usually denoted as $C_{xx,xx}, C_{xx,xy}$ etc.{cite}`McLean1967JCP,Pedersen2011JCC,Yan2020CTP`) is the preferred format for reporting. ```orca --------------------------------------------------- STATIC POLARIZABILITY TENSOR (Quadrupole/Quadrupole) --------------------------------------------------- The order in each direction is XX, YY, ZZ, XY, XZ, YZ Method : SCF Type of density : Electron Density Type of derivative : Quadrupolar Field (Direction=X) Multiplicity : 1 Irrep : 0 Relativity type : Basis : AO The raw cartesian tensor (atomic units): 60.656194448 8.024072323 8.017351959 -0.002591466 0.000000801 -0.000000184 8.024072323 55.906127614 12.837825709 -6.821368242 -0.000000954 -0.000000529 8.017351959 12.837825709 55.938851507 6.815300773 0.000000232 0.000000422 -0.002591466 -6.821368242 6.815300773 16.716647772 0.000000169 -0.000000030 0.000000801 -0.000000954 0.000000232 0.000000169 16.715850196 6.818791255 -0.000000184 -0.000000529 0.000000422 -0.000000030 6.818791255 21.534628724 diagonalized tensor: 11.893291534 13.566719080 26.357187387 46.234564137 52.663246003 76.753292120 Orientation: -0.000000018 -0.000019986 -0.000000013 -0.001433194 0.817436692 0.576016666 0.000000006 0.219691224 0.000000038 0.673107563 -0.405967809 0.577799371 0.000000008 -0.219450737 -0.000000021 -0.671194117 -0.408640606 0.578232381 -0.000000035 0.950566746 -0.000000034 -0.310519906 -0.000497156 -0.000034103 0.816443231 0.000000038 0.577425709 -0.000000037 0.000000027 0.000000000 -0.577425709 -0.000000003 0.816443231 -0.000000036 0.000000002 -0.000000003 Isotropic polarizability : 37.91138 ------------------------------------------------------------- STATIC TRACELESS POLARIZABILITY TENSOR (Quadrupole/Quadrupole) ------------------------------------------------------------- The order in each direction is XX, YY, ZZ, XY, XZ, YZ Method : SCF Type of density : Electron Density Type of derivative : Quadrupolar Field (Direction=X) Multiplicity : 1 Irrep : 0 Relativity type : Basis : AO The raw cartesian tensor (atomic units): 26.331642600 -13.160050722 -13.171591878 0.000221134 0.000000581 -0.000000065 -13.160050722 22.733889017 -9.573838294 -5.113861448 -0.000000735 -0.000000324 -13.171591878 -9.573838294 22.745430172 5.113640314 0.000000154 0.000000389 0.000221134 -5.113861448 5.113640314 12.537485829 0.000000127 -0.000000022 0.000000581 -0.000000735 0.000000154 0.000000127 12.536887647 5.114093442 -0.000000065 -0.000000324 0.000000389 -0.000000022 5.114093442 16.150971543 ``` :::{note} - **Like the quadrupole moments themselves, the dipole-quadrupole and quadrupole-quadrupole polarizabilities depend on the gauge origin of the `%elprop` module. The latter can be changed using the `Origin` keyword in `%elprop`; see section {ref}`sec:spectroscopyproperties.elprop`.** ::: At the SCF level, dynamic (frequency-dependent) dipole polarizabilities can be computed using either purely real or purely imaginary frequencies. ```orca %elprop polar 1 freq_r 0.08 # purely real frequencies #freq_i 0.08 # purely imaginary frequencies end ``` In the example above, the dynamic dipole polarizability tensor for a single real frequency of 0.8 a.u. is computed. For every frequency, linear response equations must be solved for all component of the perturbation operator (3 Cartesian components of the electric dipole). Note that imaginary-frequency polarizabilities are computed with the same costs as real-frequency polarizabilities. By a simple contour integration they can be used to compute dispersion coefficients. For methods that do not support analytic polarizabilities, one can calculate numeric dipole-dipole and quadrupole-quadrupole polarizabilities, either by finite differentiation of dipole/quadrupole moments with respect to a finite dipole/quadrupole electric field, or by double finite differentiation of the total energy with respect to a finite dipole/quadrupole electric field. The latter can be done using compound scripts (see {ref}`sec:workflowsautomatization.compound`, {ref}`sec:workflowsautomatization.compound_examples`). At the MP2 level, polarizabilities can currently be calculated analytically using the RI ({ref}`sec:modelchemistries.mp2.rimp2.response`) or DLPNO ({ref}`sec:modelchemistries.mp2.dlpnoresp`) approximations or in all-electron canonical calculations, but for canonical MP2 with frozen core orbitals the dipole moment has to be differentiated numerically in order to obtain the polarizability tensor. In general in such cases, you should keep in mind that tight SCF convergence is necessary in order to not get too much numerical noise in the second derivative. Also, you should experiment with the finite field increment in the numerical differentiation process. As an example, the following Compound job can be used to calculate the seminumeric polarizability at the MP2 level (replacing the now deprecated usage of `Polar 2`): ```{literalinclude} ../../orca_working_input/ORCA_Test_0798.inp :language: orca ``` with the file `semiNumericalPolarizability.cmp` containing: ```{literalinclude} ../../orca_working_input/semiNumericalPolarizability.cmp :language: orca ``` For more details on Compound jobs in general, see {ref}`sec:workflowsautomatization.compound`. For other correlation methods, where not even relaxed densities are available, only a fully numeric approach (using compounds scripts) is possible and requires obnoxiously tight convergence. Note that polarizability calculations have higher demands on basis sets. A rather nice basis set for this property is the Sadlej one (see {ref}`sec:essentialelements.basisset.builtin`). In relation to electric properties, it might be interesting to know that it is possible to carry out finite electric field calculations in ORCA. See section {ref}`sec:essentialelements.finEfield` for more information on such calculations. The static hyperpolarizability tensor is calculated as an analytical third derivative of the energy with respect to three dipole perturbations. It is currently available for SCF methods (HF, DFT). For example, consider the input: ```orcainput ! RHF def2-SVP TightSCF %elprop hyperpol 1 end *xyz 0 1 C 0.000000 -0.000000 -0.528012 O -0.000000 0.000000 0.650528 H -0.000000 0.931690 -1.116318 H -0.000000 -0.931690 -1.116318 * ``` The hyperpolarizability can be found printed in the output as ```orca --------------------------------------------------- STATIC HYPERPOLARIZABILITY TENSOR --------------------------------------------------- Method : SCF Type of density : Electron Density Type of derivative : Electric Field (Direction=X) Multiplicity : 1 Irrep : 0 Basis : AO The raw Cartesian tensor (atomic units): ( x x x ): -0.000000 ( x x y ): -0.000000 ( x x z ): 7.076938 ( x y x ): -0.000000 ( x y y ): -0.000000 ( x y z ): -0.000000 ( x z x ): 7.076938 ( x z y ): -0.000000 ( x z z ): -0.000000 ( y x x ): -0.000000 ( y x y ): -0.000000 ( y x z ): -0.000000 ( y y x ): -0.000000 ( y y y ): 0.000000 ( y y z ): 49.371503 ( y z x ): -0.000000 ( y z y ): 49.371503 ( y z z ): 0.000000 ( z x x ): 7.076938 ( z x y ): -0.000000 ( z x z ): -0.000000 ( z y x ): -0.000000 ( z y y ): 49.371503 ( z y z ): 0.000000 ( z z x ): -0.000000 ( z z y ): 0.000000 ( z z z ): 44.368713 ``` (sec:spectroscopyproperties.atomicelectric)= ## Atomic Dipole Moment and Polarizabilities ```{index} Atomic Dipole Moment and Polarizabilities ``` ORCA includes a mechanism by which electric properties can be decomposed into atomic contributions. In a nutshell, the method relies on the partitioning of real space into atomic regions. These regions are defined by a smooth weighting function for atom A $$f_{A}\left( \mathbf{r} \right) = \frac{\varrho_{A}^{(0)}(\mathbf{r})}{\sum_{B}^{}{\varrho_{B}^{(0)}(\mathbf{r})}}$$ The quantities $\varrho_{A}^{(0)}(r)$ are spherically symmetric atomic densities (expanded in a reasonably large set of Gaussian s-functions) for the neutral atoms that exit pretabulated in the code. Using these partitioning functions, an atomic multipole moment is defined as: $$Q_{l}^{m}(A) = \sum_{\mu\nu}P_{\mu\nu}\int{\mu(\mathbf{r}){\widehat{q}}_{l}^{m}\nu(\mathbf{r})f_{A}\left( \mathbf{r} \right)d\mathbf{r}}$$ Here **P** is the density matrix and ${\hat{q}}_{l}^{m}$ the operator for a multipole moment of order l with component m. For example, for the dipole moment l=1 and m=0,+1,-1. Likewise, for a quadrupole moment l=2, m=2,1,0,-1,-2 and so on. The partitioned integration over space is carried out by numerical integration. In the limit of exact numerical integration, the construction guarantees that the atomic moments sum to the overall multipole moment. Because numerical integration introduces a small error, the accumulated moments $$Q_{l}^{m} = \sum_{A}^{}{Q_{l}^{m}(A)}$$ Will show small deviation from the analytically integrated moments that the program also calculates prints. Clearly, the same strategy can be used for response properties. For example, for polarizabilities, one decomposes it into the atomic components as follows: $$\alpha_{KL}(A) = - \sum_{\mu\nu}\frac{\partial P_{\mu\nu}}{\partial F_{K}}\int{\mu(\mathbf{r})\mathbf{r}_{L}\nu(\mathbf{r})f_{A}\left( \mathbf{r} \right)d\mathbf{r}}$$ Here **F** is an external electric field and the derivative of the density matrix is the response density with respect to this field. In order to trigger this calculation use: ```orca %elprop Polar true PolarAtom true Dipole true DipoleAtom true Quadrupole true QuadrupoleAtom true end ``` :::{note} - Atomic electric properties are available presently at the SCF level for dipole moments, quadrupole moments and polarizabilities - The dipole moment is well-known to be gauge dependent for charged molecules. Hence, the way the dipole moment is decomposed here, the atomic dipole moments refer to the global origin and there is a gauge dependence for charged atoms. Consequently, one needs to proceed with some caution in the interpretation of these local dipole moments. ::: PAPER: - Grigorash, D.; Müller, S.; Heid, E.; Neese, F.; Liakos, D.; Riplinger, C.; Garcia-Rates, M.; Paricaud, P. Erling H.S.; Smirnova, I.; Yan, W. A comprehensive approach to incorporating intermolecular dispersion into the openCOSMO-RS model. Part 2: Atomic polarizabilities, 2025, submitted (sec:spectroscopyproperties.properties.elprop.keywords)= ## Elprop Keywords ```{index} Electrical Properties Keywords ``` Calculation of first order (electric dipole and quadrupole moments) and second order (polarizabilities) electric properties can be requested in the `%elprop` input block. The second order properties can be calculated through the solution of the CP-SCF (see {ref}`sec:essentialelements.cpscf`) or CP-CASSCF (see {ref}`sec:spectroscopyproperties.casscfresp`) equations. Details are shown below: ```orca %elprop Dipole true Quadrupole true Polar true PolarVelocity true # polarizability w.r.t. velocity perturbations PolarDipQuad true # dipole-quadrupole polarizability PolarQuadQuad true # quadrupole-quadrupole polarizability Hyperpol true # the dip/dip/dip hyperpolarizability freq_r 0.00 # purely real frequency (default: static calculation) freq_i 0.00 # purely imaginary frequency (default: static calculation) Solver CG # CG(conjugate gradient) # other options: DIIS or POPLE(default) MaxDIIS 5 # max. dimension of DIIS method Shift 0.2 # level shift used in DIIS solver Tol 1e-3 # Convergence of the CP-SCF equations # (norm of the residual) MaxIter 64 # max. number of iterations in CPSCF PrintLevel 2 Origin CenterOfElCharge # center of electronic charge CenterOfNucCharge # center of nuclear charge CenterOfSpinDens # center of spin density CenterOfMass # center of mass (default) N # position of atom N (starting at 0) X,Y,Z # explicit position of the origin # in coordinate input units (Angstrom by default) end ``` The most efficient and accurate way to calculate the polarizability analytically is to use the coupled-perturbed SCF method. The most time consuming and least accurate way is the numerical second derivative of the total energy. Note that the numerical differentiation requires: (a) tightly or even very tightly converged SCF calculations and (b) carefully chosen field increments. If the field increment is too large then the truncation error will be large and the values will be unreliable. On the other hand, if the field increment is too small the numerical error associated with the finite difference differentiation will get unacceptably large up to the point where the whole calculation becomes useless.