Title: ETH_Sem5 / AdvQC / 10_Atoms_Molecules

Q: What is the principle approach to many-eletron atoms?
A: - As for one-electron atoms: central-field problem after Born-Oppenheimer approximation (infinitely heavy nucleus at center of coordinate system)
- Exploit spherical coordinates to obtain ordinary differential equations in the radial coordinate only. Can be solved with numerical discretization techniques on a mesh of grid points (see lecture Quantum Chemistry).
- Take radial one-electron Dirac Hamiltonian from Dirac hydrogen atom and Coulomb electron-electron interaction operator in spherical coordinates (see lecture Quantum Chemistry) to obtain the many-electron Hamiltonian
\[H_\mathrm{el}= h^D(r_i) - \bm{1}_4 m_ec^2 + \sum_{j<i}^N\sum_{\nu=0}^\infty \frac{r^\nu_{<}}{r^{\nu+1}_{>}}\frac{4\pi}{2\nu+1}\sum_{m=-\nu}^\nu Y_{\nu m}^\star(\vartheta_i,\varphi_i)Y_{\nu m}(\vartheta_i,\varphi_i)\]
- Ansatz for atomic spinor separated into radial and spherical parts:
\[\Psi_i(\bm{r}) \rightarrow \Psi_{n_i \kappa_i m_j(i)}(\bm{r})=\frac{1}{r}\begin{pmatrix} P_{n_i \kappa_i}(r) \chi_{\kappa_i m_{j(i)}}(\vartheta, \varphi) \\\mathrm{i} Q_{n_i \kappa_i}(r) \chi_{-\kappa_i m_{j(i)}}(\vartheta, \varphi) \end{pmatrix}\]
- The only unknowns are the radial functions \(P_{n_i \kappa_i}(r)\) and \(Q_{n_i \kappa_i}(r)\)! For them, we need to write the SCF equations in the radial coordinate only.

Q: Sketch the spinor-energies of the bound states for a Schrödinger many-electron atom and a Dirac many-electron atom!
A: \(\bullet\) The radial functions for different \(j\) quantum numbers are not the same anymore.
\(\rightarrow\) Spin-orbit coupling manifests itself by differently shaped radial
functions at high \(Z\), e.g., when going from Kr (\(Z=36\)) to Rn (\(Z=86\)) At small \(Z\) the radial functions of, e.g., \(np_{1/2}\) and \(np_{3/2}\) can hardly be distinguished.
\(\bullet\) The small component is nucleus-centered and hence turns out to be atomic in molecules, which can be exploited in computational solution procedures.
\(\bullet\) Compared to the large components, the small components are the ``smaller'' the smaller the nuclear charge number \(Z\) is.

Q: How can we employ a Gaussian basis set expansion to the 4-spinors?
A: Taking advantage of the \(2\times2\)-superstructure of the one-electron Dirac Hamiltonian, an expansion in terms of 2-spinor basis functions \(\phi_\mu^L\left(\bm{r}_A\right)\) and \(\phi_\mu^S\left(\bm{r}_A\right)\) can be done where we might use spherical two-component basis functions (\(\chi_{\kappa_{\mu} m_{\mu}}(\vartheta_A, \varphi_A)\) are the Pauli spinors).
\[\psi_i(\bm{r})=\begin{pmatrix} \sum_\mu c_{i \mu}^{(L)} \phi_\mu^L\left(\bm{r}_A\right) \\ \sum_\mu c_{i \mu}^{(S)} \phi_\mu^S\left(\bm{r}_A\right) \end{pmatrix}\]
\[\begin{aligned} \phi_\mu^L\left(\bm{r}_A\right) &= \frac{P_{\mu}(r_A)}{r_A} \chi_{\kappa_{\mu} m_{\mu}}(\vartheta_A, \varphi_A) \\ \phi_\mu^S\left(\bm{r}_A\right) &= \imag\frac{Q_{\mu}(r_A)}{r_A} \chi_{-\kappa_{\mu} m_{\mu}}(\vartheta_A, \varphi_A) \end{aligned}\]
with \(\psi_i\) being a molecular spinor, and expand the unknown radial functions either into (i) an atomic Gaussian basis set
\[\begin{aligned} &P_\mu^{\mathrm{GTO}}\left(r_A\right)=N_\mu^L r_A^{l_\mu+1} \exp \left(-\zeta_\mu r_A^2\right) \\ &Q_\mu^{\mathrm{GTO}}\left(r_A\right)=N_\mu^S\left[\left(\kappa_\mu+l_\mu+1\right)-\zeta_\mu r_A\right] r_A^{l_\mu} \exp \left(-\zeta_\mu r_A^2\right) \end{aligned}\]
or into (ii) Cartesian Gaussian functions,
\[\begin{aligned} &P_\mu^{\mathrm{cGTO}}\left(\bm{r}_A\right)=N_\mu^L x^{\alpha_\mu} y^{\beta_\mu} z^{\gamma_\mu} \exp \left(-\zeta_\mu r_A^2\right) \\ &Q_\mu^{\operatorname{cGTO}}\left(\bm{r}_A\right)=N_\mu^S x^{\alpha_\mu} y^{\beta_\mu} z^{\gamma_\mu} \exp \left(-\zeta_\mu r_A^2\right) \end{aligned}\]
The subscript \(A\) denotes that the electronic coordinate is to be taken relative to the atomic nucleus \(A\) at which it is centered, \(\pos_A = \pos - \bm{R}_A\).

Q: What has to be taken care of when performing the basis set expansion? What is the consequence?
A: The approximate kinetic balance condition,
\[\psi_i^S = \frac{c}{2m_ec^2 - W + \epsilon_i} (\bm{\sigma}\cdot\bm{p})\psi^L\approx\frac{\bm{\sigma}\cdot\bm{p}}{2m_ec}\psi^L\]
is to be fulfilled \(\Rightarrow\) first derivatives of Cartesian primitive Gaussians,
\[\pdv{x}x^{\alpha_\mu}\exp(-\zeta_\mu r_A^2) = \left[ \alpha_\mu x^{\alpha_\mu-1}- 2 \zeta_\mu x^{\alpha_\mu+1} \right]\exp(-\zeta_\mu r_A^2)\]
with \(r_A^2 = \left(x^{\alpha_\mu}\right)^2 + \left(y^{\alpha_\mu}\right)^2 + \left(z^{\alpha_\mu}\right)^2\), have to be evaluated to associate the exponents \(\zeta_\mu^S\) for the small component with those of the large component \(\zeta_\mu^L\).
Because of the derivative operator \(\mom = -\imag\hbar\nabla\) every large component Gaussian basis spinor \(\phi_i^L\) gives rise to two small-component basis
functions with the same exponent.
As a consequence, the small-component basis will contain functions with higher angular momentum than the large-component basis and can become almost twice as large as the large-component basis.

Q: How can the usage of different basis sets for the large and small component be circumvented?
A: If we rewrite the SCF equations for a given molecular spinor \(\psi_i\) in split notation as
\[\begin{pmatrix} W & c\bm{\sigma}\cdot\bm{p} \\ c\bm{\sigma}\cdot\bm{p} & W-2m_ec^2 \end{pmatrix} \begin{pmatrix} \psi_i^L \\ \psi_i^S \end{pmatrix} = \epsilon_i \begin{pmatrix} \psi_i^L \\ \psi_i^S \end{pmatrix}\]
where \(W\) again collects \(V_\mathrm{nuc}\) and the electron-electron mean-field
potential energy terms, we may insert the approximate kinetic balance
condition and obtain
\[\begin{pmatrix} W & c\bm{\sigma}\cdot\bm{p} \\ c\bm{\sigma}\cdot\bm{p} & W-2m_ec^2 \end{pmatrix} \begin{pmatrix} \psi_i^L \\ \frac{\bm{\sigma}\cdot\bm{p}}{2m_ec}\tilde{\psi}_i^L \end{pmatrix} = \epsilon_i \begin{pmatrix} \psi_i^L \\ \frac{\bm{\sigma}\cdot\bm{p}}{2m_ec}\tilde{\psi}_i^L \end{pmatrix}\]
Now, we absorb the \((c\bm{\sigma}\cdot\bm{p})\)-dependent prefactor into the operator
\[\begin{pmatrix} W & c\bm{\sigma}\cdot\bm{p}\frac{\bm{\sigma}\cdot\bm{p}}{2m_ec} \\ c\bm{\sigma}\cdot\bm{p} & (W-2m_ec^2)\frac{\bm{\sigma}\cdot\bm{p}}{2m_ec} \end{pmatrix} \begin{pmatrix} \psi_i^L \\ \tilde{\psi}_i^L \end{pmatrix} = \epsilon_i \begin{pmatrix} \psi_i^L \\ \frac{\bm{\sigma}\cdot\bm{p}}{2m_ec}\tilde{\psi}_i^L \end{pmatrix}\]
and use the same basis sets for upper and lower components (the tilde indicates the approximate nature of the kinetic balance).
Note that the remaining prefactor on the right-hand side would then later
be absorbed in a modified overlap matrix.
This equation is closely related to the modified Dirac equation.

Q: Write down the Fock operator and the Dirac-Hartree-Fock equations in split notation! What is the general procedure to derive the Hartree-Fock equations?
A: - Set up an expression for the Hartree-Fock energy which depends explicitly on the variational parameters (i.e., either molecular orbitals or basis functions).
- Set up a Lagrangian functional \(\mathcal{L}\) which incorporates the conditions which were used when deriving the energy expression (orthonormality of the MOs).
- Apply the variational principle by setting the infinitesimal variation of the Lagrangian functional to zero, \(\delta\mathcal{L} = 0\). From the resulting equations, the Hartree-Fock equations can be derived.
\(\begin{aligned} \displaystyle \text{Fock operator: } \quad f=\left(\begin{array}{cc} V & c \bm{\sigma} \cdot \bm{p} \\ c \bm{\sigma} \cdot \bm{p} & \left(V-2 m c^2\right) \end{array}\right) \\ +\sum_{j=1}^N\left[\left(\begin{array}{cc} j_j & 0 \\ 0 & j_j \end{array}\right)-\left(\begin{array}{cc} k_j^{L L} & k_j^{S L} \\ k_j^{L S} & k_j^{S S} \end{array}\right)\right] \end{aligned}\)
\[\begin{aligned} j_j\ket{\psi_i^L} &= \left(j_j^{LL}+j_j^{SS}\right)\ket{\psi_i^L} \\ &= \left(\braket{g}{\psi_j^L} + \braket{g}{\psi_j^S}\right)\ket{\psi_i^L} \\ &= \left(\int \frac{1}{r_{12}} \rho(2) \mathrm{d} 2 \right)\ket{\psi_i^L} \\ k_j^{LS}\ket{\psi_i^L} &= \braket{\psi_j^L|g|\psi_i^L}\ket{\psi_j^S} \\ k_j^{SS}\ket{\psi_i^L} &= \braket{\psi_j^S|g|\psi_i^L}\ket{\psi_j^S} \end{aligned}\]
\(\displaystyle \text{Dirac-Hartree-Fock equations: } \quad f\left(\begin{array}{c} \psi_i^L \\ \psi_i^S \end{array}\right) =\sum_j \epsilon_{i j}\left(\begin{array}{c} \psi_j^L \\ \psi_j^S \end{array}\right) \)
The four-dimensional Fock operator is given by
\[f = h^D + V\]
with \(h^D\) being the field-free Dirac Hamiltonian and all potential terms are collected in
\[V = V_\mathrm{nuc} + V_{ee}\]
where \(V_\mathrm{nuc}\) contains the point-like or finite-nucleus attraction of an electron with all nuclei in the system and \(V_{ee}\) comprises all electron-electron interaction terms whose explicit form depends on the choice of the approximation to the many-electron wave function (e.g. Dirac-Coulomb or Dirac-Coulomb-Breit). While \(h^D\) also included \(V_\mathrm{nuc}\) in case of the Dirac hydrogen atom, it is now taken as two separate operators (i.e., \(h^D\) is the field-free Dirac Hamiltonian) for later convenience.

Q: Write down the Dirac-Hartree-Fock-Roothaan equations!
A: \[ \begin{aligned}&\left(\begin{array}{c} \psi_i^L \\ \psi_i^S \end{array}\right)=\left(\begin{array}{c} \sum_{a=1}^{K_L} c_{i a}^L \phi_a^L \\ \sum_{b=1}^{K_S} c_{i b}^S \phi_b^S \end{array}\right) \ \\ &\Rightarrow\ \underbrace{ \left(\begin{array}{ll} \bm{F}^{L L} & \bm{F}^{S L} \\ \bm{F}^{L S} & \bm{F}^{S S}\\ \end{array}\right)\left(\begin{array}{c} \bm{c}_{\bm{i}}^L\\ \bm{c}_{\bm{i}}^S \end{array}\right)=\epsilon_i\left(\begin{array}{cc} \bm{S}^{L L} & 0 \\ 0 & \bm{S}^{S S} \\ \end{array}\right)\left(\begin{array}{c} \bm{c}_{\bm{i}}^L \\ \bm{c}_i^S \end{array}\right)}_{\displaystyle \bm{F c}=\bm{S c \epsilon}} \end{aligned}\]
with the Fock matrix
\[\begin{array}{>{\displaystyle}l >{\displaystyle}l} \bm{F}^{L L} =\bm{V}^{L L}+\bm{J}^{L L}-\bm{K}^{L L} & \\ \bm{F}^{S L} = c[\bm{\sigma} \cdot \bm{p}]^{S L}-\bm{K}^{S L} \\ \bm{F}^{L S} = c[\bm{\sigma} \cdot \bm{p}]^{L S}-\bm{K}^{L S} & \\ \bm{F}^{S S} =\bm{V}^{S S}-2 m_e c^2 \bm{S}^{S S}+\bm{J}^{S S}-\bm{K}^{S S} \end{array}\]
\[\begin{aligned} \left(\bm{K}^{XX}\right)_{ab} & =\sum_{\mu \nu} D_{\nu \mu}^{XX}\left\langle\phi_a^X(1) \phi_\nu^X(2)\left|\frac{1}{r_{12}}\right| \phi_\mu^X(1) \phi_b^X(2)\right\rangle \\ \left(\bm{K}^{XY}\right)_{a b} & =\sum_{\mu \nu} D_{\nu \mu}^{XY}\left\langle\phi_a^Y(1) \phi_\nu^X(2)\left|\frac{1}{r_{12}}\right| \phi_\mu^Y(1) \phi_b^X(2)\right\rangle \\ \left(\bm{J}^{X X}\right)_{a b} & =\sum_{\mu \nu}\left[D_{\nu \mu}^{L L}\left\langle\phi_a^X(1) \phi_\nu^L(2)\left|\frac{1}{r_{12}}\right| \phi_b^X(1) \phi_\mu^L(2)\right\rangle\right. \\ & \quad +\left.D_{\nu \mu}^{S S}\left\langle\phi_a^X(1) \phi_\nu^S(2)\left|\frac{1}{r_{12}}\right| \phi_b^X(1) \phi_\mu^S(2)\right\rangle\right] \\ \left(\bm{J}^{X Y}\right)_{a b} & =0 \end{aligned}\]