TEXT   63

one electron operators

Guest on 19th May 2022 03:28:57 PM

  1. :orphan:
  2.  
  3.  
  4. .. _one_electron_operators:
  5.  
  6. One-electron operators
  7. ======================
  8.  
  9. Syntax for the specification of one-electron operators
  10. ------------------------------------------------------
  11.  
  12. A general one-electron property operator in 4-component calculations is generated from
  13. linear combinations of the basic form:
  14.  
  15. .. math::
  16.  
  17.   \hat{P} = f M_{4 \times 4} \hat{\Omega}
  18.  
  19. with the scalar factor :math:`f` and the scalar operator :math:`\hat{\Omega}`, and where
  20.  
  21. .. math::
  22.  
  23.   M_{4 \times 4}
  24.  
  25. is one of the following :math:`4 \times 4` matrices:
  26.  
  27. .. math::
  28.  
  29.   I_{4 \times 4}, \gamma_5, \beta \gamma_5,
  30.  
  31.   i\alpha_x, i\alpha_y, i\alpha_z
  32.  
  33.   \Sigma_x, \Sigma_y,  \Sigma_z
  34.  
  35.   \beta \Sigma_x, \beta \Sigma_y, \beta \Sigma_z
  36.  
  37.   i \beta \alpha_x, i \beta \alpha_y, i \beta \alpha_z
  38.  
  39. 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
  40. quaternion symmetry scheme of DIRAC (see :cite:`Saue1999` and :cite:`Salek2005` for more information).
  41.  
  42. Operator types
  43. --------------
  44.  
  45. There are 21 basic operator types used in DIRAC, listed in this Table:
  46.  
  47. ===========  ============================================================================= ===============
  48. **Keyword**  **Operator form**                                                             **Nr. factors**
  49. ===========  ============================================================================= ===============
  50. DIAGONAL     :math:`f I_{4 \times 4} \Omega`                                               1
  51. XALPHA       :math:`f \alpha_x \Omega`                                                     1
  52. YALPHA       :math:`f \alpha_y \Omega`                                                     1
  53. ZALPHA       :math:`f \alpha_z \Omega`                                                     1
  54. XAVECTOR     :math:`f_1 \alpha_y \Omega_z - f_2 \alpha_z \Omega_y`                         2
  55. YAVECTOR     :math:`f_1 \alpha_z \Omega_x - f_2 \alpha_x \Omega_z`                         2
  56. ZAVECTOR     :math:`f_1 \alpha_x \Omega_y - f_2 \alpha_y \Omega_x`                         2
  57. ALPHADOT     :math:`f_1 \alpha_x \Omega_x + f_2 \alpha_y \Omega_y + f_3 \alpha_z \Omega_z` 3
  58. GAMMA5       :math:`f \gamma_5 \Omega`                                                     1
  59. XSIGMA       :math:`f \Sigma_x \Omega`                                                     1
  60. YSIGMA       :math:`f \Sigma_y \Omega`                                                     1
  61. ZSIGMA       :math:`f \Sigma_z \Omega`                                                     1
  62. XBETASIG     :math:`f \beta \Sigma_x \Omega`                                               1
  63. YBETASIG     :math:`f \beta \Sigma_y \Omega`                                               1
  64. ZBETASIG     :math:`f \beta \Sigma_z \Omega`                                               1
  65. XiBETAAL     :math:`f i \beta \alpha_x \Omega`                                             1
  66. YiBETAAL     :math:`f i \beta \alpha_y \Omega`                                             1
  67. ZiBETAAL     :math:`f i \beta \alpha_z \Omega`                                             1
  68. BETA         :math:`f \beta \Omega`                                                        1
  69. SIGMADOT     :math:`f_1 \Sigma_x \Omega_x + f_2 \Sigma_y \Omega_y + f_3 \Sigma_z \Omega_z` 1
  70. iBETAGAMMA5  :math:`f i \beta \gamma_5 \Omega`                                             1
  71. ===========  ============================================================================= ===============
  72.  
  73.  
  74. Operator specification
  75. ----------------------
  76.  
  77. Operators are specified by the keyword :ref:`HAMILTONIAN_.OPERATOR` with the following
  78. arguments::
  79.  
  80.   .OPERATOR
  81.    'operator name'
  82.    operator type keyword
  83.    operator labels for each component
  84.    FACTORS
  85.    factors for each component
  86.    CMULT
  87.    COMFACTOR
  88.    common factor for all components
  89.  
  90. Note that the arguments following the keyword :ref:`HAMILTONIAN_.OPERATOR` must start with
  91. a blank. The arguments are optional, except for the operator label.
  92. Component factors as well as the common factor are all one if not specified.
  93.  
  94.  
  95. List of one-electron operators
  96. ------------------------------
  97.  
  98. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  99. | **Operator**| **Description**                  | **Symmetry**   | **Components** | **Operators**                    |
  100. | **label**   |                                  |                |                |                                  |
  101. +=============+==================================+================+================+==================================+
  102. | MOLFIELD    | Nuclear attraction integrals     | Symmetric      | MOLFIELD       | :math:`\Omega_1 = \sum_K V_{iK}` |
  103. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  104. | OVERLAP     | Overlap integrals                | Symmetric      | OVERLAP        | :math:`\Omega_1 = 1`             |
  105. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  106. | BETAMAT     | Overlap integrals, only SS-block | Symmetric      | BETAMAT        | :math:`\Omega_1 = 1`             |
  107. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  108. | DIPLEN      | Dipole length integrals          | Symmetric      | XDIPLEN        | :math:`\Omega_1 = x`             |
  109. |             |                                  |                +----------------+----------------------------------+
  110. |             |                                  |                | YDIPLEN        | :math:`\Omega_2 = y`             |
  111. |             |                                  |                +----------------+----------------------------------+
  112. |             |                                  |                | ZDIPLEN        | :math:`\Omega_3 = z`             |
  113. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  114. | DIPVEL      | Dipole velocity integrals        | Anti-symmetric | XDIPVEL        |                                  |
  115. |             |                                  |                +----------------+----------------------------------+
  116. |             |                                  |                | YDIPVEL        |                                  |
  117. |             |                                  |                +----------------+----------------------------------+
  118. |             |                                  |                | ZDIPVEL        |                                  |
  119. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  120. | QUADRUP     | Quadrupole moments integrals     | Symmetric      | XXQUADRU       |                                  |
  121. |             |                                  |                +----------------+----------------------------------+
  122. |             |                                  |                | XYQUADRU       |                                  |
  123. |             |                                  |                +----------------+----------------------------------+
  124. |             |                                  |                | XZQUADRU       |                                  |
  125. |             |                                  |                +----------------+----------------------------------+
  126. |             |                                  |                | YYQUADRU       |                                  |
  127. |             |                                  |                +----------------+----------------------------------+
  128. |             |                                  |                | YZQUADRU       |                                  |
  129. |             |                                  |                +----------------+----------------------------------+
  130. |             |                                  |                | ZZQUADRU       |                                  |
  131. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  132. | SPNORB      | Spatial spin-orbit integrals     | Anti-symmetric | X1SPNORB       |                                  |
  133. |             |                                  |                +----------------+----------------------------------+
  134. |             |                                  |                | Y1SPNORB       |                                  |
  135. |             |                                  |                +----------------+----------------------------------+
  136. |             |                                  |                | Z1SPNORB       |                                  |
  137. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  138. | SECMOM      | Second moments integrals         | Symmetric      | XXSECMOM       | :math:`\Omega_1 = xx`            |
  139. |             |                                  |                +----------------+----------------------------------+
  140. |             |                                  |                | XYSECMOM       | :math:`\Omega_2 = xy`            |
  141. |             |                                  |                +----------------+----------------------------------+
  142. |             |                                  |                | XZSECMOM       | :math:`\Omega_3 = xz`            |
  143. |             |                                  |                +----------------+----------------------------------+
  144. |             |                                  |                | YYSECMOM       | :math:`\Omega_4 = yy`            |
  145. |             |                                  |                +----------------+----------------------------------+
  146. |             |                                  |                | YZSECMOM       | :math:`\Omega_5 = yz`            |
  147. |             |                                  |                +----------------+----------------------------------+
  148. |             |                                  |                | ZZSECMOM       | :math:`\Omega_6 = zz`            |
  149. +-------------+----------------------------------+----------------+----------------+----------------------------------+
  150.  
  151.  
  152. =========== ====================================================================================================================
  153. **Keyword** **Description**
  154. =========== ====================================================================================================================
  155. THETA       Traceless theta quadrupole integrals
  156. CARMOM      Cartesian moments integrals, symmetric integrals, (l + 1)(l + 2)/2 components ( l = i + j + k)
  157. SPHMOM      Spherical moments integrals (real combinations), symmetric integrals, (2l + 1) components ( m = +0, -1, +1, ..., +l)
  158. SOLVENT     Electronic solvent integrals
  159. FERMI C     One-electron Fermi contact integrals
  160. PSO         Paramagnetic spin-orbit integrals
  161. SPIN-DI     Spin-dipole integrals
  162. DSO         Diamagnetic spin-orbit integrals
  163. SDFC        Spin-dipole + Fermi contact integrals
  164. HDO         Half-derivative overlap integrals
  165. S1MAG       Second order contribution from overlap matrix to magnetic properties
  166. ANGLON      Angular momentum around the nuclei
  167. ANGMOM      Electronic angular momentum around the origin
  168. LONMOM      London orbital contribution to angular momentum
  169. MAGMOM      One-electron contributions to magnetic moment
  170. KINENER     Electronic kinetic energy
  171. DSUSNOL     Diamagnetic susceptibility without London contribution
  172. DSUSLH      Angular London orbital contribution to diamagnetic susceptibility
  173. DIASUS      Angular London orbital contribution to diamagnetic susceptibility
  174. NUCSNLO     Nuclear shielding integrals without London orbital contribution
  175. NUCSLO      London orbital contribution to nuclear shielding tensor integrals
  176. NUCSHI      Nuclear shielding tensor integrals
  177. NEFIELD     Electric field at the individual nuclei
  178. ELFGRDC     Electric field gradient at the individual nuclei, cartesian
  179. ELFGRDS     Electric field gradient at the individual nuclei, spherical
  180. S1MAGL      Bra-differentiation of overlap matrix with respect to magnetic field
  181. S1MAGR      Ket-differentiation of overlap matrix with respect to magnetic field
  182. HDOBR       Ket-differentiation of HDO-integrals with respect to magnetic field
  183. NUCPOT      Potential energy at the nuclei
  184. HBDO        Half B-differentiated overlap matrix
  185. SQHDO       Half-derivative overlap integrals not to be antisymmetrized
  186. DSUSCGO     Diamagnetic susceptibility with common gauge origin
  187. NSTCGO      Nuclear shielding integrals with common gauge origin
  188. EXPIKR      Cosine and sine integrals
  189. MASSVEL     Mass velocity integrals
  190. DARWIN      Darwin type integrals
  191. CM1         First order magnetic field derivatives of electric field
  192. CM2         Second order magnetic field derivatives of electric field
  193. SQHDOR      Half-derivative overlap integrals not to be anti-symmetrized
  194. SQOVLAP     Second order derivatives overlap integrals
  195. =========== ====================================================================================================================
  196.  
  197.  
  198. Examples of using various operators
  199. -----------------------------------
  200.  
  201. We give here several concrete examples on how to construct operators for
  202. various properties.
  203.  
  204. Kinetic part of the Dirac Hamiltonian
  205. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  206.  
  207. The kinetic part of the Dirac Hamiltonian may be specified by::
  208.  
  209.   .OPERATOR
  210.    'Kin energy'
  211.    ALPHADOT
  212.    XDIPVEL
  213.    YDIPVEL
  214.    ZDIPVEL
  215.    COMFACTOR
  216.    -68.51799475
  217.  
  218. where -68.51799475 is :math:`-c/2`.
  219.  
  220. The speed of light :math:`c` is an important parameter in relativistic
  221. theory, but its explicit value in atomic units not necessarily remembered.
  222. A simpler way to specify the kinetic energy operator is therefore::
  223.  
  224.  
  225.   .OPERATOR
  226.    'Kin energy'
  227.    ALPHADOT
  228.    XDIPVEL
  229.    YDIPVEL
  230.    ZDIPVEL
  231.    CMULT
  232.    COMFACTOR
  233.    -0.5
  234.  
  235. where the keyword *CMULT* assures multiplication of the common factor -0.5 by :math:`c`.
  236. This option has the further advantage that *CMULT* follows any user-specified modification
  237. of the speed of light, as provided by :ref:`GENERAL_.CVALUE`.
  238.  
  239. XAVECTOR
  240. ~~~~~~~~
  241.  
  242. Another example::
  243.  
  244.   .OPERATOR
  245.    'B_x'
  246.    XAVECTOR
  247.    ZDIPLEN
  248.    YDIPLEN
  249.    CMULT
  250.    COMFACTOR
  251.    -0.5
  252.  
  253.  
  254. The program will assume all operators to be Hermitian and will therefore insert
  255. an imaginary phase *i* if necessary (applies to antisymmetric scalar
  256. operators).
  257.  
  258. If no other arguments are given, the program assumes the operator to be a
  259. diagonal operator and expects the operator name to be the component label, for
  260. instance::
  261.  
  262.   .OPERATOR
  263.    OVERLAP
  264.  
  265.  
  266. Dipole moment as finite field perturbation
  267. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  268.  
  269. Another example is the finite perturbation calculation with the :math:`\hat{z}`
  270. dipole length operator added to the Hamiltonian (don't forget to decrease the
  271. symmetry of your system):
  272.  
  273. .. math::
  274.  
  275.     \hat{H} = \hat{H}_0 + 0.01 \cdot \hat{z}
  276.  
  277. ::
  278.  
  279.   .OPERATOR
  280.    ZDIPLEN
  281.    COMFACTOR
  282.    0.01
  283.  
  284.  
  285. Fermi-contact integrals
  286. ~~~~~~~~~~~~~~~~~~~~~~~
  287.  
  288. Here is an example where the Fermi-contact (FC) integrals for a certain nucleus
  289. are added to the Hamiltonian in a finite-field calculation.  Let's assume you
  290. are looking at a PbX dimer (order in the .mol file: 1.  Pb, 2. X) and you want
  291. to add to the Dirac-Coulomb :ref:`**HAMILTONIAN` the FC integrals for the Pb
  292. nucleus as a perturbation with a given field-strength (FACTORS).
  293.  
  294. **Important note:** The raw density values obtained after the fit of
  295. your finite-field energies need to be scaled by
  296. :math:`\frac{1}{\frac{4 \pi g_{e}}{3}} = \frac{1}{8.3872954891254192}`,
  297. a factor that originates from the definition of the operator for
  298. calculating the density at the nucleus::
  299.  
  300.   **HAMILTONIAN
  301.   .OPERATOR
  302.    'Density at nucleus'
  303.    DIAGONAL
  304.    'FC Pb 01'
  305.    FACTORS
  306.    -0.000000001
  307.  
  308. Here is next example of how-to calculate the electron density at the
  309. nucleus as an expectation value :math:`\langle 0 \vert \delta(r-R) \vert 0
  310. \rangle` for a Dirac-Coulomb HF wave function including a decomposition of the
  311. molecular orbital contributions to the density::
  312.  
  313.   **DIRAC
  314.   .WAVE FUNCTION
  315.   .PROPERTIES
  316.   **HAMILTONIAN
  317.   **WAVE FUNCTION
  318.   .SCF
  319.   **PROPERTIES
  320.   .RHONUC
  321.   *EXPECTATION
  322.   .ORBANA
  323.   *END OF
  324.  
  325.  
  326. Cartesian moment expectation value
  327. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  328.  
  329. In the following example I am calculating a cartesian moment expectation value
  330. :math:`\langle 0 \vert x^1 y^2 z^3 \vert 0 \rangle` for a Levy-Leblond HF wave
  331. function::
  332.  
  333.   **DIRAC
  334.   .WAVE FUNCTION
  335.   .PROPERTIES
  336.   **HAMILTONIAN
  337.   .LEVY-LEBLOND
  338.   **WAVE FUNCTION
  339.   .SCF
  340.   **PROPERTIES
  341.   *EXPECTATION
  342.   .OPERATOR
  343.    CM010203
  344.   *END OF

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