Hartree-Fock, HF

The APMO/HF level wave function for a multispecies molecular system in the ground state, \(\Psi_0\), is constructed as a product of a single-configurational wave function, \(\Phi_{\alpha}\), :

\[{\Psi}_0= \prod_{\alpha}^{N_{sp}} \Phi_{\alpha}\]

In the most common case, \(\Phi_{\alpha}\) is built as a single slater determinant of single-particle spin molecular orbitals \(\chi_i\) as

\[\begin{split}\begin{aligned} \Psi_{\mathrm{SD}}(\overline{\mathbf{x}})=\frac{1}{\sqrt{N_{\alpha} !}}\left|\begin{array}{cccc} \chi_1\left(\mathbf{x}_1\right) & \chi_1\left(\mathbf{x}_2\right) & \ldots & \chi_1\left(\mathbf{x}_{N_{\alpha}}\right) \\ \chi_2\left(\mathbf{x}_1\right) & \chi_2\left(\mathbf{x}_2\right) & \ldots & \chi_2\left(\mathbf{x}_{N_{\alpha}}\right) \\ \ldots & \ldots & \ldots & \ldots \\ \chi_{N_{\alpha}}\left(\mathbf{x}_1\right) & \chi_{N_f}\left(\mathbf{x}_2\right) & \ldots & \chi_{N_{\alpha}}\left(\mathbf{x}_{N_{\alpha}}\right) \end{array}\right|, \label{chap2:eq:SD}\end{aligned}\end{split}\]

here \(\alpha\) symbol is used to represent quantum species, \(\mathbf{x}\) corresponds to their spin-coordinates vector, and \(\overline{\mathbf{x}}\) is the full set of 4\(N\) coordinates. The spatial part of these spin orbitals, \(\phi\), is written as a linear combination of atomic orbitals \(\varphi\) with a basis set of size \(B\) (notice that the number of atomic centers can be greater than the number of classical nuclei \(N_c\)) as

\[\begin{aligned} \phi_i^{\alpha}(\mathbf{r}_i) = \sum_{\mu}^{N_{bas}^{\alpha}} C_{\mu}^{\alpha} \varphi_{\mu}^{\alpha}(\mathbf{r}_i;\mathbf{R}_{\mu}) \end{aligned}\]

Commonly these atomic orbitals are constructed with GTFs that depend on the spatial coordinate of one single particle \(r_i\) and are centered on a fixed position \(R\) for each atomic center. See Basis

The molecular orbitals are obtained by solving the coupled Fock equations, see SCF,

\[\begin{aligned} \label{chap2:eq:FockEquation} f^\alpha(i)\phi^\alpha_i=\varepsilon^\alpha_i\phi^\alpha_i, \end{aligned}\]

where \(\varepsilon_i\) are the single particle orbital energies. The effective one-particle Fock operators, \(f^\alpha(i)\), for the quantum species \(e^-\) and \(e^+\) are expanded as

\[\begin{aligned} \label{chap2:eq:FockOperator} f^{\alpha}(i)=h^{\alpha}(i) + \sum^{N_{\alpha}}_{j} (q^{\alpha})^2 [J^{\alpha}_j - K^{\alpha}_j] + \sum_{\beta\ne\alpha}^{N_{sp}} \sum^{N_{\beta}}_{j} q^{\alpha}q^{\beta} J^{\beta}_j. \end{aligned}\]

In the above equation \(h^\alpha(i)\) is the single-particle core Hamiltonian

\[\begin{aligned} \label{chap2:eq:CoreHamiltonian} h^\alpha(i)=-\frac{1}{2m_{\alpha}}\nabla_{i}^{2} + \sum_{J}^{N_c}\frac{q^{\alpha} Z_J }{R_{iJ}}, \end{aligned}\]

and \(J^\alpha\) and \(K^\alpha\) are the Coulomb and exchange operators defined as

\[\begin{aligned} \label{chap2:eq:CoulombOperator} J^\alpha_j(1)\phi^\alpha_i(1)= q^\alpha q^\alpha\left[\int d{\bf r}_2\phi^{\alpha*}_j(2) \frac{1}{r_{12}} \phi^{\alpha}_j(2)\right]\phi^\alpha_i(1) ,\end{aligned}\]

\[\begin{aligned} \label{chap2:eq:ExchangeOperator} K^{\alpha} _j(1)\phi^{\alpha}_i(1)=\left[\int d{\bf r}_2\phi^{{\alpha}*}_j(2)\frac{1}{r_{12}} \phi^{\alpha}_i(2)\right]\phi^{\alpha}_j(1) .\end{aligned}\]

In addition, \(J^\beta\) is the operator which accounts for the Coulomb potential between particles of different quantum species, thus is the term which couples the electronic and positronic Fock equations, and is defined as

\[\begin{aligned} \label{chap2:eq:CoulombCouplingOperator} J^\beta_j(1)\phi^\alpha_i(1)= q^\beta q^\alpha\left[\int d{\bf r}_2\phi^{\beta*}_j(2) \frac{1}{r_{12}} \phi^{\beta}_j(2)\right]\phi^\alpha_i(1) . \end{aligned}\]

In openLOWDIN, these expressions are implemented in a matrix form [1]

\[\begin{split}\begin{aligned} \begin{aligned} S_{\mu \nu}^\alpha & =\int d r_1 \varphi_\mu^\alpha(1) \varphi_\nu^\alpha(1) \\ F_{\mu \nu}^\alpha & =H_{\mu \nu}^\alpha+q^{\alpha} q^{\alpha} G_{\mu \nu}^\alpha + \sum_{\beta\ne\alpha}^{N_{sp}}q^{\alpha} q^{\beta} G_{\mu \nu}^\beta \\ H_{\mu \nu}^\alpha & =\int d r_1 \varphi_\mu^\alpha(1) h^\alpha(1) \varphi_\nu^\alpha(1) \\ G_{\mu \nu}^\alpha & =\sum_{\lambda \sigma} P_{\lambda \sigma}^\alpha\left[\left(\mu^\alpha \nu^\alpha \mid \sigma^\alpha \lambda^\alpha\right)- (1/2) \left(\mu^\alpha \lambda^\alpha \mid \sigma^\alpha \nu^\alpha\right)\right] \\ G_{\mu \nu}^\beta & =\sum_{\lambda \sigma} P_{\lambda \sigma}^\beta\left(\mu^\alpha \nu^\alpha \mid \sigma^\beta \lambda^\beta\right), \end{aligned}\end{aligned}\end{split}\]

where \(S_{\mu \nu}\), \(F_{\mu \nu}\), \(H_{\mu \nu}\), and \(G_{\mu \nu}\) correspond to the overlap, one-core Hamiltonian, and two-particles matrices, which all run over the total number of atomic orbitals \(\mu,\nu\). Here, the chemistry notation of two-particles integrals \(( \varphi_{\mu} \varphi_{\nu} | \varphi_{\sigma} \varphi_{\lambda} )\) has been simplified to \(( \mu \nu | \sigma \lambda)\). See Integrals.

In addition, the density matrix elements are defined from the coefficient matrix \(\mathbf{C}\) of the molecular orbitals expansion and the fermionic orbital occupation \(\eta\)

\[P_{\mu \nu}^{\alpha} = \eta^{\alpha}\sum_{\lambda}^{occ^\alpha} C^{\alpha}_{\mu \lambda} C^{* \alpha}_{\lambda \nu}.\]

These coefficient matrices are found by diagonalizing the Roothan-Hall equations

\[\mathbf{F C} = \epsilon \mathbf{S C}. \label{chap2:eq:roothan}\]

Finally, the total Hartree-Fock energy is computed from

\[\begin{aligned} E_0=\frac{1}{2} \sum_{\alpha}^{N_{sp}} \sum_{\mu\nu}^{occ^{\alpha}} P_{\mu \nu}^\alpha\left(H_{\mu \nu}^\alpha+F_{\mu \nu}^\alpha\right). \label{chap2:eq:HF}\end{aligned}\]

• frozen= [character] Default "NONE"

• freezeNonElectronicOrbitals= [logical] Default .false.

• freezeElectronicOrbitals= [logical] Default .false.

• hartreeProductGuess= [logical] Default .false.

• readCoefficients= [logical] Default .true.

• readFchk= [logical] Default .false.

• writeCoefficientsInBinary= [logical] Default .true.

• readEigenvalues= [logical] Default .false.

• readEigenvaluesInBinary= [logical] Default .true.

• writeEigenvaluesInBinary= [logical] Default .true.

• noSCF= [logical] Default .false.

• finiteMassCorrection= [logical] Default .false.

• removeTranslationalContamination= [logical] Default .false.

• buildTwoParticlesMatrixForOneParticle= [logical] Default .false.

• buildMixedDensityMatrix= [logical] Default .false.

• onlyElectronicEffect= [logical] Default .false.

• electronicWaveFunctionAnalysis= [logical] Default .false.

• isOpenShell= [logical] Default .false.

• getGradients= [logical] Default .false.

• HFprintEigenvalues= [logical] Default .true.

• HFprintEigenvectors= [character] Default "OCCUPIED"

• overlapEigenThreshold= [float] Default 1.0E-8_8

• electricField(:)= [float] Default 0.0_8

• multipoleOrder= [integer] Default 0