Title: ETH_Sem5 / AdvQC / 12_Elimination_Techniques_Pauli_Breit-Pauli_ZORA

Q: What is the central idea of all elimination methods? Where does one arrive at if no approximations are used?
A: The central idea of all elimination methods is to exploit the kinetic-balance condition \(\psi^S=X \psi^L\), for the substitution of \(\psi^S\) by an expression for the large component only.
This yields a two-component eigenvalue equation for \(\psi^L\),
\[(V+c \bm{\sigma} \cdot \bm{p} X) \psi^L=\epsilon \psi^L\]
Historically, the energy-dependent \(X\)-operator has been employed,
\[X=\frac{c \bm{\sigma} \cdot \bm{p}}{2 m_e c^2-V+\epsilon}\]
which yields the two-component equation
\[(V-\epsilon) \psi^L+\frac{1}{2 m_e c^2}[(c \bm{\sigma} \cdot \bm{p}) \omega(c \bm{\sigma} \cdot \bm{p})] \psi^L=0\]
with
\[\omega=\omega(\epsilon)=\left[1-\frac{V-\epsilon}{2 m_e c^2}\right]^{-1}\]
So far, no approximation has been introduced, and the two-component equation still yields the exact Dirac eigenvalues and large components. However, the \(X\)-operator depends on the desired solutions, \(\epsilon\), itself!

Q: What is the Pauli Hamiltonian/Elimination? How can the Hamiltonian be derived?
A: The Pauli approximation utilizes the properties of the geometric series in
order to convert \(\omega\) into \(\omega = \sum_{k=0}^\infty \left(\frac{V-\epsilon}{2 m_e c^2}\right)^k\). Inserting the first two terms with \(k=0,1\) into the analytical two-component equation from before yields a kinetic energy operator of the form
\[\frac{1}{2 m_e c^2}\left[(c \bm{\sigma} \cdot \bm{p})\left(1+\frac{V-\epsilon}{2 m_e c^2}\right)(c \bm{\sigma} \cdot \bm{p})\right]=\frac{\bm{p}^2}{2 m_e}+\frac{\bm{\sigma} \cdot \bm{p} V \bm{\sigma} \cdot \bm{p}}{4 m_e^2 c^2}-\frac{\epsilon \bm{p}^2}{4 m_e^2 c^2}\]
We can apply Dirac's relation and rewrite the middle term:
\[\begin{aligned} \bm{\sigma} \cdot \bm{p} V \bm{\sigma} \cdot \bm{p}=\bm{p} V \bm{p}+\mathrm{i} \bm{\sigma} \cdot(\bm{p} V \times \bm{p}) \\ \bm{p} V\bm{p}=\frac{1}{2}\left[V \bm{p}^2+\bm{p}^2 V\right]+\frac{\hbar^2}{2}(\Delta V)\end{aligned}\]
By recalling the TISE and assuming \(\epsilon^{nr}\approx\epsilon\) , we may write the anticommutator:
\[\begin{aligned} V \Psi&=\left(\epsilon^{n r}-\frac{\bm{p}^2}{2 m_e}\right) \Psi\\ & \thus \frac{1}{2}\left[V \bm{p}^2+\bm{p}^2 V\right] \approx \epsilon^{n r} \bm{p}^2-\frac{\bm{p}^4}{2 m_e} \end{aligned}\]
This yields the relativistic kinematic corrections to the Pauli Hamiltonian (see lecture on Dirac equation),
\[h^{\text {Pauli }}=\frac{\bm{p}^2}{2 m_e}+V-\frac{\bm{p}^4}{8 m_e^3 c^2}+\frac{\hbar^2}{8 m_e^2 c^2}(\Delta V)+\frac{\hbar}{4 m_e^2 c^2} \bm{\sigma} \cdot[(\nabla V) \times \bm{p}]\]

Q: Which terms does the Pauli Hamiltonian include?
A: The Pauli Hamiltonian reads
\[h^{\text {Pauli }}=\overbrace{\underbrace{\frac{\bm{p}^2}{2 m_e}+V}_{h^{S}}}^{\text{nonrelativistic}} \overbrace{\underbrace{-\frac{\bm{p}^4}{8 m_e^3 c^2}}_\text{mass-velocity operator}+ \underbrace{\frac{\hbar^2}{8 m_e^2 c^2}(\Delta V)}_\text{Darwin term}+\underbrace{\frac{\hbar}{4 m_e^2 c^2} \bm{\sigma} \cdot[(\nabla V) \times \bm{p}]}_\text{spin-orbit coupling term}}^\text{relativistic correction terms}\]
These operators describe all relativistic corrections up to \(\mathcal{O}\left(c^{-2}\right)\). The spin-orbit term can be cast in a more convenient form for a hydrogen-like atom with a point-like nucleus (see exercise):
\[h^{\mathrm{SO}}=\frac{\hbar}{4 m_e^2 c^2} \bm{\sigma} \cdot[(\nabla V) \times \bm{p}]=\frac{Z e^2}{2 m_e^2 c^2} \frac{\bm{s} \cdot \bm{l}}{r^3}\]
The Darwin operator for hydrogen-like atoms is often written in a different
form (see exercise):
\[h^{\text {Darwin }}=\frac{\hbar^2}{8 m_e^2 c^2}(\Delta V)=\frac{\hbar^2 \pi Z e^2}{2 m_e^2 c^2} \delta^{(3)}(\bm{r})\]
One can see that the Darwin operator is related to the electron density!

Q: What is the problem of the Pauli Hamiltonian/Elimination?
A: Major problem of the Pauli Hamiltonian: It is a \(1/c\)-expansion and therefore cannot be used in variational procedures.
The minus sign of the mass-velocity term leads to a strongly attractive potential for states with high momentum, and would therefore lead to variational collapse, since it is not bounded from below. This insurmountable problem cannot be remedied by going to higher orders, since the geometric series expansion
\[\omega = \sum_{k=0}^\infty \left(\frac{V-\epsilon}{2 m_e c^2}\right)^k\]
is only valid for \(\abs{V-\epsilon}<2m_ec^2\), which is certainly violated in regions close to the nucleus.
From a formal point of view, in a momentum-space representation arbitrarily-high
momentum basis functions will contribute to every bound state so that a \(1/c\) expansion will not represent the original Hamiltonian (see the FW discussion).
Also the Darwin term produces difficulties as it degenerates to the singular Delta
distribution term in the case of a point-like nucleus.
Therefore the Pauli Hamiltonian and all other operators based on simple expansions of \(\omega\) in powers of \(c^{-2}\) according to the expansion above are in general singular and ill-defined, and must not be used for variational procedures (cf. FW transformation)!

Q: What is the Breit-Pauli Hamiltonian/Elimination?
A: In the Breit-Pauli Hamiltonian/Elimination, the two-electron operators are considered as well! The most convenient way to reduce the Breit equation to an essentially two-dimensional Pauli-like equation is to apply a unitary transformation as in FW. We apply the fpFW transformation to the Breit operator, with \(U_0(W_{[0]})=a_{0,0} \mathbf{1}+\sum_{k=1}^{\infty} a_{0, k} W_{[0]}^k\) and \(\Psi = \Psi(\bm{r}_1,\bm{r}_2)\),
\[\left\{U _ { 0 } \left(h_1^D+h_2^D+\frac{q_1 q_2}{r_{12}}-\frac{q_1 q_2}{2}\left[\frac{\bm{\alpha}_1 \cdot \bm{\alpha}_2}{r_{12}}+\frac{\left(\bm{r}_{12} \cdot \bm{\alpha}_1\right)\left(\bm{r}_{12} \cdot \bm{\alpha}_2\right)}{r_{12}^3}\right]\right) U_0^{\dagger}\right\}\left(U_0 \Psi\right)=E\left(U_0 \Psi\right)\]
For two particles we have
\[U_0 = U_0(1)\otimes U_0(2)\qquad\qquad\text{e.g., } U_0 = \exp{W_{[0]}(1)}\otimes\exp{W_{[0]}(2)}\]
These \(W_{[0]}\) operators operate only on the corresponding coordinates. Hence,
\[U_0 h_1^D U_0^{\dagger}=U_0(1) h_1^D U_0^{\dagger}(1) \otimes \underbrace{U_0(2) U_0^{\dagger}(2)}_{=1} \longrightarrow\left[U_0(1) h_1^D U_0^{\dagger}(1)\right]^{L L} \approx h^{\text {Pauli }}(1)\]
and vice versa for 2. Choosing \(a_{0,0} = a_{0,1} = 1\) the total unitary transformation is
\[U_0=\mathbf{1}+W_{[0]}(1)+W_{[0]}(2)+W_{[0]}(1) W_{[0]}(2)+\frac{1}{2} W_{[0]}^2(1)+\frac{1}{2} W_{[0]}^2(2)+O\left(c^{-3}\right)\]

Q: How is the two-electron interaction transformed in the Breit-Pauli Hamiltonian?
A: The unitary transformation of the two electron interaction \(g(1,2) = V_C(1,2)+B_0(1,2)\) with \(U_0\) reads
\[\begin{aligned} &U_0 g(1,2) U_0^{\dagger} \approx\left(\mathbf{1}+W_{[0]}(1)+W_{[0]}(2)+W_{[0]}(1) W_{[0]}(2)+\frac{1}{2} W_{[0]}^2(1) + \frac{1}{2} W_{[0]}^2(2)\right) \\ &\hphantom{=}\times \left[V_C(1,2)+B_0(1,2)\right]\left(\mathbf{1}-W_{[0]}(1)-W_{[0]}(2)+W_{[0]}(1) W_{[0]}(2)+\frac{1}{2} W_{[0]}^2(1)+\frac{1}{2} W_{[0]}^2(2)\right) \end{aligned}\]
which produces terms of the structure \(\mathcal{E}(1)\mathcal{E}(2)\), \(\mathcal{O}(1)\mathcal{O}(2)\), \(\mathcal{E}(1)\mathcal{O}(2)\) and \(\mathcal{O}(1)\mathcal{E}(2)\), denoting that the corresponding term features an even/odd structure w.r.t. particle 1/2.
All-Even Operators: We now identify all-even operators and write in compact form
\[\begin{aligned} \left[U_0 g(1,2) U_0^{\dagger}\right]_{\mathcal{E}} &= V_C(1,2)+\frac{1}{2}\left[W_{[0]}(1),\left[W_{[0]}(1), V_C(1,2)\right]\right] \\ &\hphantom{,}+ \frac{1}{2}\left[W_{[0]}(2),\left[W_{[0]}(2), V_C(1,2)\right]\right] + \left[W_{[0]}(1),\left[W_{[0]}(2), B_0(1,2)\right]\right]+\mathcal{O}\left(c^{-3}\right) \\ &\equiv V_C(1,2)+\widetilde{V}_C(1,2)+\widetilde{B}_0(1,2) \end{aligned}\]
To arrive at the two-component external-field-free Breit-Pauli Hamiltonian to be used in perturbation theory
\[H_\mathrm{BP}=H_0+H_1+H_2+H_3+H_4+H_5\]

Q: Interpret the interactions in each term of the Breit-Pauli Hamiltonian: \normalsize{\[ \begin{aligned} H_0 &= \sum_{i=1}^2\left[\frac{\bm{p}_i^2}{2 m_e}+V_\mathrm{nuc}(i)\right]+\frac{e^2}{r_{12}}=\sum_{i=1}^2 h^S(i)+\frac{e^2}{r_{12}} \\ H_1 &= -\sum_{i=1}^2 \frac{\bm{p}_i^4}{8 m_e^3 c^2} \\ H_2 &= -\frac{e^2}{2 m_e^2 c^2}\left[\frac{\bm{p}_1 \cdot \bm{p}_2}{r_{12}}+\frac{\left(\bm{r}_{12} \cdot \bm{p}_1\right)\left(\bm{r}_{12} \cdot \bm{p}_2\right)}{r_{12}^3}\right] \\ H_3 &= \frac{e}{2 m_e^2 c^2}\left[\left(\bm{E}_1 \times \bm{p}_1\right) \cdot \bm{s}_1+\frac{2 e}{r_{12}^3}\left(\bm{r}_{12} \times \bm{p}_2\right) \cdot \bm{s}_1 + \left(\bm{E}_2 \times \bm{p}_2\right) \cdot \bm{s}_2+\frac{2 e}{r_{12}^3}\left(\bm{r}_{21} \times \bm{p}_1\right) \cdot \bm{s}_2\right] \\ H_4 &= -\frac{\mathrm{i} e \hbar}{4 m_e^2 c^2} \sum_{i=1}^2 \bm{p}_i \cdot \bm{E}_i = \frac{e \hbar^2}{4 m_e^2 c^2} \sum_{i=1}^2 \frac{ \nabla^2_i}{{q_e}}V(i) \\ H_5 &= \frac{e^2}{m_e^2 c^2}\left[-\frac{8 \pi}{3}\left(\bm{s}_1 \cdot \bm{s}_2\right) \delta^{(3)}\left(\bm{r}_{12}\right) + \frac{\bm{s}_1 \cdot \bm{s}_2}{r_{12}^3}-3 \frac{\left(\bm{r}_{12} \cdot \bm{s}_1\right)\left(\bm{r}_{12} \cdot \bm{s}_2\right)}{r_{12}^5}\right] \end{aligned}\]
A: - \(H_0\) recovers the unperturbed non-relativistic many-electron Hamiltonian
- \(H_1\) is the so-called mass-velocity term and is a purely scalar contribution which arises from the variation of the mass of the electron with its speed
- \(H_2\) exactly resembles the classical Darwin energy, while the Breit operator required only velocity rather than momentum operators. It encounters retardation effects due to the electromagnetic field generated by an electron
- \(H_3\) is the spin-orbit coupling term which contains the spin-same-orbit coupling given by the vector product of the electric field with the momentum operator as
\[\begin{aligned} &\left(\bm{E}_1 \times \bm{p}_1\right) \cdot \bm{s}_1 \\ &= \left(-\nabla_1\frac{-e}{r_{12}} \times \bm{p}_1\right) \cdot \bm{s}_1 \\ &= e\left(-\frac{\bm{r_{12}}}{r_{12}^3} \times \bm{p}_1\right) \cdot \bm{s}_1 \\ &= -\frac{e}{r_{12}^3} \left[\bm{l}\cdot\bm{s}_1\right]\end{aligned}\]
Further, we have a spin-other-orbit coupling term hidden in the rest of \(H_3\).
\[ \frac{e}{2 m_e^2 c^2}\left[\frac{2 e}{r_{12}^3}\left(\bm{r}_{12} \times \bm{p}_2\right) \cdot \bm{s}_1 +\frac{2 e}{r_{12}^3}\left(\bm{r}_{21} \times \bm{p}_1\right) \cdot \bm{s}_2\right] \]
- \(H_4\) produces the Laplacian of the potential, i.e. the Darwin correction
- \(H_5\) contains the Gaunt-like spin-spin interaction namely a dipole-dipole (last two terms) and a Fermi-contact-type interaction (first term)

Q: What is the Breit-Pauli Hamiltonian useful for?
A: The Breit-Pauli Hamiltonian is widely applied in the molecular response theory of properties; e.g., for magnetic resonance spectroscopy: The splitting of states due to the interaction of all internal angular momenta is called zero-field splitting. For electron spin resonance spectroscopy a phenomenological spin Hamiltonian is employed to describe these splittings of states;
\[H_{\mathrm{eff}}^{(0)}=\bm{S} \cdot \bm{D} \cdot \bm{S}\]
An explicit expression for \(\bm{D}\) in the elementary two-spin problem can be deduced, of course, from the explicit expression of \(H_5\).
The spin-orbit part of the Breit-Pauli Hamiltonian can be subdivided into
a sum of one-electron terms \(h_{\mathrm{BP}}^{\mathrm{SO}}\) and a sum of two-electron terms \(g_{\mathrm{BP}}^{\mathrm{SO}}\) stemming from the Breit interaction
\[h_{\mathrm{BP}}^{\mathrm{SO}}=\frac{1}{2 m^2 c^2} \sum_i \sum_I \frac{Z_I\left(\bm{r}_{i I} \times \bm{p}_i\right) \cdot \bm{s}_i}{r_{i I}^3}\]
\[g_{\mathrm{BP}}^{\mathrm{SO}}=\underbrace{-\frac{1}{2 m^2 c^2} \sum_{i, j \neq i} \frac{\left(\bm{r}_{i j} \times \bm{p}_i\right) \cdot \bm{s}_i}{r_{i j}^3}}_{g_{\mathrm{BP}}^{\mathrm{SSO}}}+\underbrace{\frac{1}{m^2 c^2} \sum_{i, j \neq i} \frac{\left(\bm{r}_{i j} \times \bm{p}_i\right) \cdot \bm{s}_j}{r_{i j}^3}}_{g_{\mathrm{BP}}^{\mathrm{SOO}}}\]
which all result, when operator \(\mathrm{H}_3\) is evaluated. The two-electron operators in \(g_{\mathrm{BP}}\) have been separated into a spin-same-orbit (SSO) and spin-other-orbit (SOO) part, respectively.

Q: Explain ZORA!
A: The basic idea is to rewrite the expression for \(\omega\):
\[\begin{aligned} \omega&=\left[1-\frac{V-\epsilon}{2 m_e c^2}\right]^{-1} \\&=\frac{2 m_e c^2}{2 m_e c^2-V}\left[1+\frac{\epsilon}{2 m_e c^2-V}\right]^{-1} \\&=\frac{2 m_e c^2}{2 m_e c^2-V} \sum_{k=0}^{\infty}\left(\frac{\epsilon}{V-2 m_e c^2}\right)^k \end{aligned}\]
and to choose the new expansion parameter \(\epsilon /\left(V-2 m_e c^2\right)\), which is the starting point for the regular approximations. A truncation of this expansion for \(\omega\) defines the zero- and first-order regular approximation abbreviated as ZORA and FORA, respectively. The ZORA equation is then obtained from
\[\begin{aligned} & (V-\epsilon) \psi^L+\frac{1}{2 m_e c^2}\left[(c \bm{\sigma} \cdot \bm{p})\left\{\frac{2 m_e c^2}{2 m_e c^2-V}\right\}(c \bm{\sigma} \cdot \bm{p})\right] \psi^L=0 \\ &\thus \left[\frac{(c \bm{\sigma} \cdot \bm{p})(c \bm{\sigma} \cdot \bm{p})}{2 m_e c^2-V}+V\right] \psi^L=\epsilon \psi^L \end{aligned}\]
The ZORA Hamiltonian is also called the Chang-Pelissier-Durand Hamiltonian owing to the earlier paper by the latter authors.
Excellent agreement of orbital energies and other valence shell properties with the results from the Dirac equation is obtained, and can even be improved within the so-called scaled ZORA variant, which takes the renormalization to the transformed large component approximately into account.

Q: What is the main disadvantage of the ZORA method?
A: The main disadvantage of the method is its gauge dependence, i.e., a constant shift of the electrostatic potential does not lead to a constant shift in the energy, because the potential enters nonlinearly in the denominator of the Hamiltonian. This deficiency can, however, be approximately cured.
Of course, the construction of the ZORA kinetic energy operator requires the knowledge of the potential \( V\), which is not so straightforward to include in Fock-type operators if off-diagonal exchange contributions show up, which we have already discussed as a general complication for all reduction methods in the context of transformation schemes.
The ZORA method comes in a scalar relativistic variant, in which all spin-dependent terms have been separated, and in the fully fledged spin-orbit version.
ZORA is certainly one of the most extensively used efficient quasi-relativistic approaches in quantum chemistry.