Abstract
In the present book, we will consider molecular systems either isolated or in interaction with external electromagnetic fields. In a bottom-up approach, which we will try to follow here, a molecule, or more generally a molecular system, is regarded as a collection of electrons and nuclei in interaction with each other and possibly with external fields.
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Notes
- 1.
See the correspondence principle in Sect. 2.2.
- 2.
A bracket notation \(\langle \vert \rangle _{{\varvec{r}}}\) is used to indicate an integration over the electronic coordinates, \({{\varvec{r}}}\), alone.
- 3.
Also known as the Born expansion (see Eq. (3.56)).
- 4.
As in Eq. (3.13) a Dirac bracket notation \(\langle \vert \rangle _{{\varvec{r}}}\) has been used for the integration over the electronic coordinates, \({{\varvec{r}}}\), which have thus been left out in \(H^{mol}\), \(\Phi ^{el}_m\) and \(H^{el}\).
- 5.
In order to avoid all ambiguities and misunderstanding, it is worth noting that, in spite of its appearance, the matrix element \({T}_{n m}({{\varvec{R}}})\) is not a purely multiplicative operator (i.e. a pure number) but still contains differential operators with respect to \({{\varvec{R}}}\), acting on the nuclear functions \(\Psi _m ({{\varvec{R}}}, t)\). On the contrary, since \(H^{el} ({{\varvec{r}}}; {{\varvec{R}}})\) does not contain differential operators with respect to \({{\varvec{R}}}\), the matrix element \({V}_{n m}({{\varvec{R}}})\) is a pure multiplicative operator.
- 6.
- 7.
rve stands for rotational, vibrational, electronic.
- 8.
The second procedure may equivalently be applied to the coordinate change \(({{\varvec{R}}}_{LF}^1, \ldots , {{\varvec{R}}}_{LF}^\alpha , \ldots , {{\varvec{R}}}_{LF}^N)\) \(\rightarrow \) \(({{\varvec{R}}}_{LF}^{NCM}, {{\varvec{q}}}, {\mathbf \Theta })\) starting from the classical nuclear kinetic energy in the LF frame: \(T^{nu} = \frac{1}{2} \sum _{\alpha = 1}^N m_\alpha \dot{{\varvec{R}}}^\alpha _{LF} \cdot \dot{{\varvec{R}}}^\alpha _{LF}\).
- 9.
For diatomic molecules (\(N = 2\)), the number of internal nuclear coordinates is equal to \(3N-5 = 1\).
- 10.
Traditionally, \( m = 0, 1, 2, \ldots \) and m = 0 corresponds to the ground state.
- 11.
These transitions generally correspond to wavelengths in the ultraviolet-visible domain.
- 12.
We assume that the ground state potential energy surface has at least one local minimum.
- 13.
These transitions generally correspond to wavelengths in the infrared domain. The matrix elements can also induce transitions between rotational states corresponding to wavelengths in the microwave domain.
- 14.
See, for instance, Chap. 13 in Ref. [15].
- 15.
\(\Psi ^{0}_0 ({{\varvec{R}}})\) is an eigenstate for the Hamiltonian operator of the electronic ground state but a wavepacket for the Hamiltonian operator of the electronic state 1.
- 16.
In this section, we have dropped the subscript \({{{\varvec{r}}}}\) on brackets, as there no ambiguity: integration is performed over the electronic coordinates, \({{{\varvec{r}}}}\). In other words \(\langle \cdots \vert \cdots \rangle \) implicitly means \(\langle \cdots \vert \cdots \rangle _{{{\varvec{r}}}}\).
- 17.
Note that this spin function is an eigenfunction of the total spin operator for eigenvalue zero (singulet). The separation of an electronic wavefunction into a symmetric (antisymmetric) spatial part and a antisymmetric (symmetric) spin part is in general only possible for two-electron wavefunctions.
- 18.
If \(\frac{2 D_e}{\hbar \omega _e}\) occurs to be an integer, \(v_\text {max} = \frac{2 D_e}{\hbar \omega _e} - 1\), because \(E^{Morse}_{v_\text {max}+1} = E^{Morse}_{v_\text {max}}\), and only the first of the two seemingly-degenerate levels is physical. This is unlikely in actual cases but could happen when considering a numerical model.
- 19.
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Gatti, F., Lasorne, B., Meyer, HD., Nauts, A. (2017). Molecular Hamiltonian Operators. In: Applications of Quantum Dynamics in Chemistry. Lecture Notes in Chemistry, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-53923-2_3
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