Density-based atomistic properties

Density-based atomistic properties#

The density-based properties are computed based on the electron density of the system. We mainly focus on the cumulative multipole moments and the Mulliken population analysis. The following properties are available:

Density-based properties#

Dictionary Key

Description

Equation

p_s

Mulliken population s-shell

\(p_s = \sum_{\mu \in s \in A}^{N_{\text{BF}}}\sum_{\nu}^{N_{\text{BF}}}P_{\mu \nu} S_{\mu\nu}\)

p_p

Mulliken population p-shell

\(p_p = \sum_{\mu \in p \in A}^{N_{\text{BF}}}\sum_{\nu}^{N_{\text{BF}}}P_{\mu \nu} S_{\mu\nu}\)

p_d

Mulliken population d-shell

\(p_d = \sum_{\mu \in d \in A}^{N_{\text{BF}}}\sum_{\nu}^{N_{\text{BF}}}P_{\mu \nu} S_{\mu\nu}\)

dipm_s

dipole moment s-shell

\(\mu_s = |\boldsymbol{\mu}_s| =\sqrt{\sum_{\alpha \in \:(x,y,z)}\left( \sum_{\kappa\in s \in A}\sum_{\lambda} P_{\kappa\lambda} \left( \alpha_A S_{\lambda\kappa} - D^\alpha_{\lambda\kappa}\right) \right)^2}\)
with the dipole moment integral: \(D^\alpha_{\lambda\kappa} = \left\langle \psi_\lambda |\alpha| \psi_\kappa\right\rangle\)

dipm_p

dipole moment p-shell

\(\mu_p = |\boldsymbol{\mu}_p| =\sqrt{\sum_{\alpha \in \:(x,y,z)}\left( \sum_{\kappa\in p \in A}\sum_{\lambda} P_{\kappa\lambda} \left( \alpha_A S_{\lambda\kappa} - D^\alpha_{\lambda\kappa}\right) \right)^2}\)

dipm_d

dipole moment d-shell

\(\mu_d = |\boldsymbol{\mu}_d| =\sqrt{\sum_{\alpha \in \:(x,y,z)}\left( \sum_{\kappa\in d \in A}\sum_{\lambda} P_{\kappa\lambda} \left( \alpha_A S_{\lambda\kappa} - D^\alpha_{\lambda\kappa}\right) \right)^2}\)

qm_s

quadrupole moment s-shell

\(\Theta_s = ||\boldsymbol{\Theta}_s|| = \sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{s}-\delta_{\alpha\beta}\Theta^{\alpha\beta}_{l}\right)^2 }\) a b

qm_p

quadrupole moment p-shell

\(\Theta_p = ||\boldsymbol{\Theta}_p|| = \sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{p}-\delta_{\alpha\beta}\Theta^{\alpha\beta}_{l}\right)^2 }\) a b

qm_d

quadrupole moment d-shell

\(\Theta_d = ||\boldsymbol{\Theta}_d|| = \sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{d}-\delta_{\alpha\beta}\Theta^{\alpha\beta}_{l}\right)^2 }\) a b

q_A

atomic partial charge

\(q_A = Z'_A - \sum_{\mu\in A}^{N_{\text{BF}}} \sum_{\nu}^{N_{\text{BF}}} P_{\mu \nu} S_{\mu\nu}\)

dipm_A

atomic dipole moment

\(\mu_A = |\boldsymbol{\mu}_A| = \sqrt{\sum_{\alpha \in \:(x,y,z)}\left(\mu_A^\alpha\right)^2} \text{ with: } \mu_A^\alpha = \sum_{l\in A} \mu_l^\alpha\) with: \(l \in {s,p,d}\)

qm_A

atomic quadrupole moment

\(\Theta_A = ||\boldsymbol{\Theta}_A||=\sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{A}-\delta_{\alpha\beta}\Theta^{\alpha\beta}_{A}\right)^2 }\) with: \(\Theta_A^{\alpha\beta} = \sum_{l\in A} \Theta_l^{\alpha\beta}\)

a. with \(\Theta_l^{\alpha\beta} = \frac{3}{2} \theta_l^{\alpha\beta} - \frac{\delta_{\alpha\beta}}{2}\left( \theta_l^{xx}+ \theta_l^{yy}+\theta_l^{zz}\right)\)
and \(\theta_l^{\alpha\beta} = \sum_{\kappa\in l\in A}\sum_{\lambda} P_{\kappa\lambda} \left( \alpha_A D^\beta_{\lambda\kappa}+\beta_A D^\alpha_{\lambda\kappa} -\alpha_A\beta_A S_{\lambda\kappa} - Q^{\alpha\beta}_{\lambda\kappa}\right)\)
and the quadrupole moment integral: \(Q^{\alpha\beta}_{\lambda\kappa} = \left\langle \psi_\lambda |\alpha\beta| \psi_\kappa\right\rangle\)

  1. Kronecker delta: \(\delta_{\alpha\beta} = 1\) if \(\alpha = \beta\) and 0 otherwise.

Note

For unrestricted calculations, the properties are computed for each spin channel separately. This is currently only the case, if a spin-polarized calculation has been performed. In this case all properties are available with the suffix _alpha and _beta for the alpha and beta spin channel, respectively.

Extended density-based properties#

The extended density-based properties are computed based on the atomic properties seen above. The convolution kernel \(f_{\text{log}}(R_A,R_B)\) has been defined in the xtbml section. The following properties are available:

Density-based extended properties#

Dictionary Key

Description

Equation

ext_q_A

ext. atomic partial charge

\(q_{A,\text{ext}} = \frac{q_A}{(CN_A+1)} + \sum_{B\neq A}^{N_{\text{atoms}}} q_B \cdot \frac{f_{\text{log}(R_A,R_B)}}{(CN_B+1)}\)

ext_dipm_A

ext. atomic dipole moment

\(\mu_{A,\text{ext}} = | \boldsymbol{\mu}_{A,\text{ext}} | = \sqrt{\sum_{\alpha \in \:(x,y,z)} \left( \mu_{A,\text{ext}}^\alpha \right)^2}\) with: \(\mu_{A,\text{ext}}^\alpha = \frac{\mu_A^\alpha}{(CN_A+1)} + \sum_{B\neq A} \left( \mu_B^\alpha - \alpha_{AB} q_B\right) \frac{f_{\text{log}(R_A,R_B)}}{(CN_B+1)}\)

ext_qm_A

ext. atomic quadrupole moment

\(\Theta_{A,\text{ext}}= ||\boldsymbol{\Theta}_{A,\text{ext}}||= \sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{A,\text{ext}}-\delta_{\alpha\beta}\Theta^{\alpha\beta}_{A,ext}\right)^2 }\)
with: \(\Theta^{\alpha\beta}_{A,\text{ext}}= \frac{\Theta^{\alpha\beta}_{A}} {(CN_A+1)}+\sum_{B\neq A} \left(\Theta_B^{\alpha\beta}+\delta\Theta_B^{\alpha\beta}\right) \frac{f_{\text{log}}}{(CN_B+1)}\)
where: \(\delta\Theta_B^{\alpha\beta} = \frac{3}{2} \delta\theta_B^{\alpha\beta} - \frac{\delta_{\alpha\beta}}{2}\left( \delta\theta_B^{xx}+ \delta\theta_B^{yy}+\delta\theta_B^{zz}\right)\)
and: \(\delta\theta_B^{\alpha\beta} = -\beta_{AB}\mu^\alpha_B + \alpha_{AB}\mu^\beta_B+\alpha_{AB}\beta_{AB}q_B \text{ where: } \beta_{AB} = \beta_{A}-\beta_B\)

ext_dipm_e

ext. atomic dipole (only electronic effects)

\(\mu_{A,\text{ext},e} =|\boldsymbol{\mu}_{A,\text{ext},e}| = \sqrt{\sum_{\alpha \in \:(x,y,z)}\left(\mu_{A,\text{ext},e}^\alpha\right)^2}\)
with: \(\mu_{A,\text{ext},e}^\alpha = \frac{\mu_A^\alpha} {(CN_A+1)}+ \sum_{B\neq A} \left(\mu_B^\alpha + \alpha_{AB} p_B\right) \frac{f_{\text{log}}}{(CN_B+1)}\)

ext_qm_e

ext. atomic quadrupole (only electronic effects)

\(\Theta_{A,\text{ext},e}= ||\boldsymbol{\Theta}_{A,\text{ext},e}||= \sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{A,\text{ext},e}\right)^2-\delta_{\alpha\beta}\left(\Theta^{\alpha\beta}_{A,\text{ext},e}\right)^2 }\)
with: \(\Theta^{\alpha\beta}_{A,\text{ext},e} = \sum_{B\neq A} \left(\Theta^{\alpha\beta}_{A} +\Theta^{\alpha\beta}_{B}+\delta\Theta_{B,e}^{\alpha\beta}\right) \frac{f_{\text{log}}}{(CN_B+1)}\)
with: \(\delta\Theta_{B,e}^{\alpha\beta} = \frac{3}{2} \delta\theta_{B,e}^{\alpha\beta} - \frac{\delta_{\alpha\beta}}{2}\left( \delta\theta_{B,e}^{xx}+ \delta\theta_{B,e}^{yy}+\delta\theta_{B,e}^{zz}\right)\)
with: \(\delta\theta_{B,e}^{\alpha\beta} = -\beta_{AB}\mu^\alpha_B + \alpha_{AB}\mu^\beta_B-\alpha_{AB}\beta_{AB}p_B\)

ext_dipm_Z

ext. atomic dipole (only nuclear effects)

\(\mu_{A,\text{ext},Z} =|\boldsymbol{\mu}_{A,\text{ext},Z}| = \sqrt{\sum_{\alpha \in \:(x,y,z)}\left(\mu_{A,\text{ext},Z}^\alpha\right)^2}\)
with: \(\mu_{A,\text{ext},Z}^\alpha = \sum_{B\neq A} \left(- \alpha_{AB} Z'_B \right) \frac{f_{\text{log}}}{(CN_B+1)}\)

ext_qm_Z

ext. atomic quadrupole (only nuclear effects)

\(\Theta_{A,\text{ext},Z}= ||\boldsymbol{\Theta}_{A,\text{ext},Z}||= \sqrt{ \sum_{\alpha\beta \:\in\:(xx,xy,xz,yy,yz,zz)}2\left(\Theta^{\alpha\beta}_{A,\text{ext},Z}\right)^2-\delta_{\alpha\beta}\left(\Theta^{\alpha\beta}_{A,\text{ext},Z}\right)^2 }\)
with: \(\Theta^{\alpha\beta}_{A,\text{ext},Z} = \sum_{B\neq A} \left( \delta\Theta_{B,Z}^{\alpha\beta} \right) \frac{f_{\text{log}}}{(CN_B+1)}\)
with: \(\delta\Theta_{B,Z}^{\alpha\beta} = \frac{3}{2} \delta\theta_{B,Z}^{\alpha\beta} - \frac{\delta_{\alpha\beta}}{2}\left( \delta\theta_{B,Z}^{xx}+ \delta\theta_{B,Z}^{yy}+\delta\theta_{B,Z}^{zz}\right)\)
with: \(\delta\theta_{B,Z}^{\alpha\beta} =\alpha_{AB}\beta_{AB}Z'_B\)

Note

For unrestricted calculations, the properties are computed for each spin channel separately. This is currently only the case, if a spin-polarized calculation has been performed. In this case all properties are available with the suffix _alpha and _beta for the alpha and beta spin channel, respectively.