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:orphan: .. _one_electron_operators: One-electron operators ====================== Syntax for the specification of one-electron operators ------------------------------------------------------ A general one-electron property operator in 4-component calculations is generated from linear combinations of the basic form: .. math:: \hat{P} = f M_{4 \times 4} \hat{\Omega} with the scalar factor :math:`f` and the scalar operator :math:`\hat{\Omega}`, and where .. math:: M_{4 \times 4} is one of the following :math:`4 \times 4` matrices: .. math:: I_{4 \times 4}, \gamma_5, \beta \gamma_5, i\alpha_x, i\alpha_y, i\alpha_z \Sigma_x, \Sigma_y, \Sigma_z \beta \Sigma_x, \beta \Sigma_y, \beta \Sigma_z i \beta \alpha_x, i \beta \alpha_y, i \beta \alpha_z One thing to notice is that an imaginary :math:`i` is added to the time-antisymmetric Dirac :math:`\boldsymbol{\alpha}` - matrices and their derivatives to make them time symmetric and hence fit into the quaternion symmetry scheme of DIRAC (see :cite:`Saue1999` and :cite:`Salek2005` for more information). Operator types -------------- There are 21 basic operator types used in DIRAC, listed in this Table: =========== ============================================================================= =============== **Keyword** **Operator form** **Nr. factors** =========== ============================================================================= =============== DIAGONAL :math:`f I_{4 \times 4} \Omega` 1 XALPHA :math:`f \alpha_x \Omega` 1 YALPHA :math:`f \alpha_y \Omega` 1 ZALPHA :math:`f \alpha_z \Omega` 1 XAVECTOR :math:`f_1 \alpha_y \Omega_z - f_2 \alpha_z \Omega_y` 2 YAVECTOR :math:`f_1 \alpha_z \Omega_x - f_2 \alpha_x \Omega_z` 2 ZAVECTOR :math:`f_1 \alpha_x \Omega_y - f_2 \alpha_y \Omega_x` 2 ALPHADOT :math:`f_1 \alpha_x \Omega_x + f_2 \alpha_y \Omega_y + f_3 \alpha_z \Omega_z` 3 GAMMA5 :math:`f \gamma_5 \Omega` 1 XSIGMA :math:`f \Sigma_x \Omega` 1 YSIGMA :math:`f \Sigma_y \Omega` 1 ZSIGMA :math:`f \Sigma_z \Omega` 1 XBETASIG :math:`f \beta \Sigma_x \Omega` 1 YBETASIG :math:`f \beta \Sigma_y \Omega` 1 ZBETASIG :math:`f \beta \Sigma_z \Omega` 1 XiBETAAL :math:`f i \beta \alpha_x \Omega` 1 YiBETAAL :math:`f i \beta \alpha_y \Omega` 1 ZiBETAAL :math:`f i \beta \alpha_z \Omega` 1 BETA :math:`f \beta \Omega` 1 SIGMADOT :math:`f_1 \Sigma_x \Omega_x + f_2 \Sigma_y \Omega_y + f_3 \Sigma_z \Omega_z` 1 iBETAGAMMA5 :math:`f i \beta \gamma_5 \Omega` 1 =========== ============================================================================= =============== Operator specification ---------------------- Operators are specified by the keyword :ref:`HAMILTONIAN_.OPERATOR` with the following arguments:: .OPERATOR 'operator name' operator type keyword operator labels for each component FACTORS factors for each component CMULT COMFACTOR common factor for all components Note that the arguments following the keyword :ref:`HAMILTONIAN_.OPERATOR` must start with a blank. The arguments are optional, except for the operator label. Component factors as well as the common factor are all one if not specified. List of one-electron operators ------------------------------ +-------------+----------------------------------+----------------+----------------+----------------------------------+ | **Operator**| **Description** | **Symmetry** | **Components** | **Operators** | | **label** | | | | | +=============+==================================+================+================+==================================+ | MOLFIELD | Nuclear attraction integrals | Symmetric | MOLFIELD | :math:`\Omega_1 = \sum_K V_{iK}` | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | OVERLAP | Overlap integrals | Symmetric | OVERLAP | :math:`\Omega_1 = 1` | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | BETAMAT | Overlap integrals, only SS-block | Symmetric | BETAMAT | :math:`\Omega_1 = 1` | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | DIPLEN | Dipole length integrals | Symmetric | XDIPLEN | :math:`\Omega_1 = x` | | | | +----------------+----------------------------------+ | | | | YDIPLEN | :math:`\Omega_2 = y` | | | | +----------------+----------------------------------+ | | | | ZDIPLEN | :math:`\Omega_3 = z` | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | DIPVEL | Dipole velocity integrals | Anti-symmetric | XDIPVEL | | | | | +----------------+----------------------------------+ | | | | YDIPVEL | | | | | +----------------+----------------------------------+ | | | | ZDIPVEL | | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | QUADRUP | Quadrupole moments integrals | Symmetric | XXQUADRU | | | | | +----------------+----------------------------------+ | | | | XYQUADRU | | | | | +----------------+----------------------------------+ | | | | XZQUADRU | | | | | +----------------+----------------------------------+ | | | | YYQUADRU | | | | | +----------------+----------------------------------+ | | | | YZQUADRU | | | | | +----------------+----------------------------------+ | | | | ZZQUADRU | | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | SPNORB | Spatial spin-orbit integrals | Anti-symmetric | X1SPNORB | | | | | +----------------+----------------------------------+ | | | | Y1SPNORB | | | | | +----------------+----------------------------------+ | | | | Z1SPNORB | | +-------------+----------------------------------+----------------+----------------+----------------------------------+ | SECMOM | Second moments integrals | Symmetric | XXSECMOM | :math:`\Omega_1 = xx` | | | | +----------------+----------------------------------+ | | | | XYSECMOM | :math:`\Omega_2 = xy` | | | | +----------------+----------------------------------+ | | | | XZSECMOM | :math:`\Omega_3 = xz` | | | | +----------------+----------------------------------+ | | | | YYSECMOM | :math:`\Omega_4 = yy` | | | | +----------------+----------------------------------+ | | | | YZSECMOM | :math:`\Omega_5 = yz` | | | | +----------------+----------------------------------+ | | | | ZZSECMOM | :math:`\Omega_6 = zz` | +-------------+----------------------------------+----------------+----------------+----------------------------------+ =========== ==================================================================================================================== **Keyword** **Description** =========== ==================================================================================================================== THETA Traceless theta quadrupole integrals CARMOM Cartesian moments integrals, symmetric integrals, (l + 1)(l + 2)/2 components ( l = i + j + k) SPHMOM Spherical moments integrals (real combinations), symmetric integrals, (2l + 1) components ( m = +0, -1, +1, ..., +l) SOLVENT Electronic solvent integrals FERMI C One-electron Fermi contact integrals PSO Paramagnetic spin-orbit integrals SPIN-DI Spin-dipole integrals DSO Diamagnetic spin-orbit integrals SDFC Spin-dipole + Fermi contact integrals HDO Half-derivative overlap integrals S1MAG Second order contribution from overlap matrix to magnetic properties ANGLON Angular momentum around the nuclei ANGMOM Electronic angular momentum around the origin LONMOM London orbital contribution to angular momentum MAGMOM One-electron contributions to magnetic moment KINENER Electronic kinetic energy DSUSNOL Diamagnetic susceptibility without London contribution DSUSLH Angular London orbital contribution to diamagnetic susceptibility DIASUS Angular London orbital contribution to diamagnetic susceptibility NUCSNLO Nuclear shielding integrals without London orbital contribution NUCSLO London orbital contribution to nuclear shielding tensor integrals NUCSHI Nuclear shielding tensor integrals NEFIELD Electric field at the individual nuclei ELFGRDC Electric field gradient at the individual nuclei, cartesian ELFGRDS Electric field gradient at the individual nuclei, spherical S1MAGL Bra-differentiation of overlap matrix with respect to magnetic field S1MAGR Ket-differentiation of overlap matrix with respect to magnetic field HDOBR Ket-differentiation of HDO-integrals with respect to magnetic field NUCPOT Potential energy at the nuclei HBDO Half B-differentiated overlap matrix SQHDO Half-derivative overlap integrals not to be antisymmetrized DSUSCGO Diamagnetic susceptibility with common gauge origin NSTCGO Nuclear shielding integrals with common gauge origin EXPIKR Cosine and sine integrals MASSVEL Mass velocity integrals DARWIN Darwin type integrals CM1 First order magnetic field derivatives of electric field CM2 Second order magnetic field derivatives of electric field SQHDOR Half-derivative overlap integrals not to be anti-symmetrized SQOVLAP Second order derivatives overlap integrals =========== ==================================================================================================================== Examples of using various operators ----------------------------------- We give here several concrete examples on how to construct operators for various properties. Kinetic part of the Dirac Hamiltonian ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The kinetic part of the Dirac Hamiltonian may be specified by:: .OPERATOR 'Kin energy' ALPHADOT XDIPVEL YDIPVEL ZDIPVEL COMFACTOR -68.51799475 where -68.51799475 is :math:`-c/2`. The speed of light :math:`c` is an important parameter in relativistic theory, but its explicit value in atomic units not necessarily remembered. A simpler way to specify the kinetic energy operator is therefore:: .OPERATOR 'Kin energy' ALPHADOT XDIPVEL YDIPVEL ZDIPVEL CMULT COMFACTOR -0.5 where the keyword *CMULT* assures multiplication of the common factor -0.5 by :math:`c`. This option has the further advantage that *CMULT* follows any user-specified modification of the speed of light, as provided by :ref:`GENERAL_.CVALUE`. XAVECTOR ~~~~~~~~ Another example:: .OPERATOR 'B_x' XAVECTOR ZDIPLEN YDIPLEN CMULT COMFACTOR -0.5 The program will assume all operators to be Hermitian and will therefore insert an imaginary phase *i* if necessary (applies to antisymmetric scalar operators). If no other arguments are given, the program assumes the operator to be a diagonal operator and expects the operator name to be the component label, for instance:: .OPERATOR OVERLAP Dipole moment as finite field perturbation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Another example is the finite perturbation calculation with the :math:`\hat{z}` dipole length operator added to the Hamiltonian (don't forget to decrease the symmetry of your system): .. math:: \hat{H} = \hat{H}_0 + 0.01 \cdot \hat{z} :: .OPERATOR ZDIPLEN COMFACTOR 0.01 Fermi-contact integrals ~~~~~~~~~~~~~~~~~~~~~~~ Here is an example where the Fermi-contact (FC) integrals for a certain nucleus are added to the Hamiltonian in a finite-field calculation. Let's assume you are looking at a PbX dimer (order in the .mol file: 1. Pb, 2. X) and you want to add to the Dirac-Coulomb :ref:`**HAMILTONIAN` the FC integrals for the Pb nucleus as a perturbation with a given field-strength (FACTORS). **Important note:** The raw density values obtained after the fit of your finite-field energies need to be scaled by :math:`\frac{1}{\frac{4 \pi g_{e}}{3}} = \frac{1}{8.3872954891254192}`, a factor that originates from the definition of the operator for calculating the density at the nucleus:: **HAMILTONIAN .OPERATOR 'Density at nucleus' DIAGONAL 'FC Pb 01' FACTORS -0.000000001 Here is next example of how-to calculate the electron density at the nucleus as an expectation value :math:`\langle 0 \vert \delta(r-R) \vert 0 \rangle` for a Dirac-Coulomb HF wave function including a decomposition of the molecular orbital contributions to the density:: **DIRAC .WAVE FUNCTION .PROPERTIES **HAMILTONIAN **WAVE FUNCTION .SCF **PROPERTIES .RHONUC *EXPECTATION .ORBANA *END OF Cartesian moment expectation value ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In the following example I am calculating a cartesian moment expectation value :math:`\langle 0 \vert x^1 y^2 z^3 \vert 0 \rangle` for a Levy-Leblond HF wave function:: **DIRAC .WAVE FUNCTION .PROPERTIES **HAMILTONIAN .LEVY-LEBLOND **WAVE FUNCTION .SCF **PROPERTIES *EXPECTATION .OPERATOR CM010203 *END OF

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