Title: ETH_Sem5 / AdvQC / 13_External_Electromagnetic_Fields_Molecular_Properties

Q: How do external fields enter the Hamiltonian?
A: This has already been discussed when we introduced minimal coupling! It means that
\[ \bm{p} \longrightarrow \bm{\pi}=\bm{p}-\lambda \frac{q_e}{c} \bm{A}\]
and for the zeroth-order contribution to the 4-potential
\[ V(\lambda)=V+\lambda\left(q_e \phi\right)\]
\(V=q_e \phi\), where \(\phi\) is the scalar potential. All perturbations can be considered by simple addition of the potential energy contributions:
\[ \bm{A}=\sum_{I=1}^s \bm{A}_I \quad \text { and } \quad \phi=\sum_{I=1}^s \phi_I\]
In the following we treat the perturbation in one shot with the dimensionless order parameter \(\lambda\),
\[ \lambda \mathcal{X} \equiv \lambda\left[-\frac{q_e}{c} \bm{\alpha} \cdot \bm{A}+V\right]\]
(usually \(\lambda\) is constructed from the actually applied fields,i.e., from the length of external \(\bm{E}\) and/or \(\bm{B}\) fields, the magnetic moments \(\bm{\mu}_I\) of nuclear spin \(I\)).
It is important to understand that all perturbations are to be introduced into the Dirac Hamiltonian through minimal coupling.
All non-relativistic approximations that are known (e.g., magnetic interactions in the Breit-Pauli Hamiltonian) follow from the field-dependent Dirac-Coulomb-Breit Hamiltonian in semi-classical theory presented here!

Q: How can one calculate the associated energy shift by introducing external fields?
A: Consider one-electron operators and incorporate them in many-electron Hamiltonian
\[ H_\mathrm{el}(\lambda)=H_\mathrm{el}+\lambda \mathcal{X}=H_\mathrm{el}+\lambda \sum_{i=1}^N X(i)\]
The corresponding full stationary many-electron equation reads
\[ H_\mathrm{el}(\lambda) \Psi_A(\lambda)=E_A(\lambda) \Psi_A(\lambda)\]
where the index \(A=0,1, \ldots\) labels the different electronic states.
An external perturbation that interacts with all electrons modifies the one-electron operators
\[ H_\mathrm{el}(\lambda)=\sum_i^N\left[h^D(i)+\lambda X(i)\right]+\sum_{i<j}^N g(i, j)\]
(\(V_\mathrm{nn}\) repulsion term is suppressed).
Consequently, the first-order energy perturbation can be reduced to one-electron integrals over the spinors, \(\left\langle\psi_i|X| \psi_j\right\rangle\),
\[ E_A^{(1)}=\left\langle\Psi_A|\mathcal{X}| \Psi_A\right\rangle=\sum_{i j}^N \gamma_{i j}\left\langle\psi_i|X| \psi_j\right\rangle\]
weighted by structure factors \(\gamma_{i j}\) according to the rules for the calculation of such matrix elements.
As an example for external-field-dependent quasi-relativistic operators we provide these expressions for the Breit-Pauli Hamiltonian.
The missing external-field-dependent operators to the Breit-Pauli Hamiltonian, which is only correct up to order \(O\left(1 / c^2\right)\), are:
\[ H_6=\frac{e}{m_e c} \sum_i \bm{B}_i \cdot \bm{s}_i+\frac{e}{m_e c} \sum_i \bm{A}_i \cdot \bm{p}_i\]

Q: How can one theoretically access molecular properties?
A: Any molecular property relevant to chemistry can be understood as a perturbation of the external-field-free electronic structure!
The exact energy for any electromagnetic perturbation measured by a parameter \(\lambda\) is given by
\[E_A(\lambda)=\left\langle\Psi_A(\lambda)\left|H_\mathrm{el}(\lambda)\right| \Psi_A(\lambda)\right\rangle\]
We may consider the effect of \(\lambda\) as a small change of the unperturbed reference and thus
\[\begin{aligned} &E_A(\lambda)=E_A+\lambda E_A^{(1)}+\lambda^2 E_A^{(2)}+\ldots \\ &\Psi_A(\lambda)=\Psi_A+\lambda \Psi_A^{(1)}+\lambda^2 \Psi_A^{(2)}+\ldots \end{aligned}\]
The perturbed \(H_\mathrm{el}(\lambda)\) need not to be expanded, since it exactly depends linearly on \(\lambda\). The perturbed energy \(E_A(\lambda)\) can also be expanded into a Taylor series around \(\lambda=0\),
\[E_A(\lambda)=\sum_{k=0}^{\infty} \frac{1}{k !}\left[\frac{\mathrm{d}^k E_A(\lambda)}{\mathrm{d} \lambda^k}\right]_{\lambda=0} \lambda^k=\sum_{k=0}^{\infty} \underbrace{\frac{1}{k !}\left[E_A^{(k)}(\lambda)\right]_{\lambda=0}}_{\equiv E_A^{(k)}} \lambda^k\]
\(E_A(\lambda)\) is calculated with the Hamiltonian that considers all relevant external perturbations (i.e., additional potential energy operators) - those perturbations that are not relevant to a given problem will drop out automatically when the partial derivatives are evaluated!

Q: What are the first- and second-order corrections to the energy?
A: The lowest-order corrections for the energy are
\[\begin{aligned} &E_A^{(1)}=\left[\frac{\mathrm{d} E_A(\lambda)}{\mathrm{d} \lambda}\right]_{\lambda=0}=\left\langle\Psi_A|\mathcal{X}| \Psi_A\right\rangle \\ &E_A^{(2)}=\frac{1}{2}\left[\frac{\mathrm{d}^2 E_A(\lambda)}{\mathrm{d} \lambda^2}\right]_{\lambda=0}=\left\langle\Psi_A|\mathcal{X}| \Psi_A^{(1)}\right\rangle \end{aligned}\]
The first-order correction to the unperturbed wave function turns out to be
\[\Psi_A^{(1)}=\sum_{\substack{B=0,(B \neq A)}}^{\infty} \frac{\left\langle\Psi_B|\mathcal{X}| \Psi_A\right\rangle}{E_A-E_B} \Psi_B\]
(in practice, this perturbation of the wave function is efficiently calculated via perturbed-SCF equations).
With this perturbative correction to the wave function, we can rewrite the second-order energy expression in the familiar form as
\[E_A^{(2)}=\sum_{\substack{B=0,(B \neq A)}}^{\infty} \frac{\left\langle\Psi_A|\mathcal{X}| \Psi_B\right\rangle\left\langle\Psi_B|\mathcal{X}| \Psi_A\right\rangle}{E_A-E_B}\]

Q: Give some examples of molecular properties which can be evaluated using perturbation (response) theory! When does this method break down?
A: \[\def\arraystretch{2.5} \begin{array}{>{\displaystyle}r >{\displaystyle}c >{\displaystyle}l l} \\ \sigma_{i j}^{(A)} & = & \left(\frac{\partial^2 E_A(\lambda)}{\partial B_i \partial \mu_j^{(A)}}\right)_{B_i=\mu_j^{(A)}=0} \\ \text{ NMR shielding tensor} \\ J_{i j}^{(A B)} & = & \left(\frac{\partial^2 E_A(\lambda)}{\partial \mu_i^{(A)} \partial \mu_j^{(B)}}\right)_{\mu_i^{(A)}=\mu_j^{(B)}=0} \\ \text{ NMR spin-spin coupling} \\ g_{i j} & = & \left(\frac{\partial^2 E_A(\lambda)}{\partial B_i \partial S_j}\right)_{B_i=S_j=0} \\ \text{ ESR }g\text{-tensor} \\ A_{i j}^{(A)} & = & \left(\frac{\partial^2 E_A(\lambda)}{\partial I_i^{(A)} \partial S_j}\right)_{I_i^{(A)}=S_j=0} \\ \text{ ESR hyperfine }A\text{-tensor} \\ \xi_{i j} & = & -\left(\frac{\partial^2 E_A(\lambda)}{\partial B_i \partial B_j^{\prime}}\right)_{B_i=B_j^{\prime}=0} \\ \text{ magnetizability} \end{array}\]
(note: the additional terms in the Hamiltonian depend, of course, on the parameters involved in the differentiation!).
This description is exact in a sense that the properties will all be well defined. However, it will break down if the external electromagnetic field is no longer weak (as in strong laser fields).

Q: Why has to be taken care when operators are chosen to represent observables in approximate relativistic approaches?
A: In transformation theories such as DKH one must be aware of the so-called picture change error/effect/artefact. For instance, for the calculation of a dipole moment \(\bm{\mu}\) the simple dipole-moment operator cannot directly be employed in an expectation value with a two-component wave function; e.g.:
\[ \bm{\mu} \neq\left\langle\psi^{\mathrm{DKH}}\left|q_e \bm{r}\right| \psi^{\mathrm{DKH}}\right\rangle\]
If transformed spinors (such as DKH orbitals) are used, any observable \(O\) needs to be transformed accordingly:
\[\begin{aligned} \langle O\rangle &= \left\langle\psi\left|U^{\dagger} U O U^{\dagger} U\right| \psi\right\rangle \\ &=\left\langle\tilde{\psi}\left|U O U^{\dagger}\right| \tilde{\psi}\right\rangle \\ &=\left\langle\tilde{\psi}^L\left|\tilde{O}^{L L}\right| \tilde{\psi}^L\right\rangle \end{aligned}\]
Occassionally, in two-component quantum chemical calculations of expectation values the quantity
\[ \bar{O} \equiv\left\langle\tilde{\psi}^L\left|O^{L L}\right| \tilde{\psi}^L\right\rangle\]
is calculated instead; the change of picture for operator \(O\) is neglected.
As a consequence of this erroneous proceeding, the artifical difference
\[ \operatorname{PCE}(O)=\langle O\rangle-\bar{O}\]
is the picture change error of the transition from the four-component to
the two-component representation. For many properties, this error is negligible. But it can be very large (e.g., for contact densities as needed, for instance, in Mössbauer spectroscopy).

Q: How can one evaluate the electron density from DKH orbitals?
A: Sum of squared transformed orbitals does not yield electron density!
\(\Rightarrow\) The density operator must be transformed first!