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2017 | Supplement | Buchkapitel

3. Molecular Hamiltonian Operators

verfasst von : Fabien Gatti, Benjamin Lasorne, Hans-Dieter Meyer, André Nauts

Erschienen in: Applications of Quantum Dynamics in Chemistry

Verlag: Springer International Publishing

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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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Vorheriges Kapitel Quantum Mechanical Background

Nächstes Kapitel Vibronic Couplings

See the correspondence principle in Sect. 2.2.

A bracket notation \(\langle \vert \rangle _{{\varvec{r}}}\) is used to indicate an integration over the electronic coordinates, \({{\varvec{r}}}\), alone.

Also known as the Born expansion (see Eq. (3.56)).

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 \(H^{el}\).

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.

Equations (3.29) and (3.30) clearly show that the nuclear wavefunction \(\Psi _m^{ad} ({{\varvec{R}}}, t)\) depends on the choice of the electronic basis functions, in particular here the adiabatic electronic basis set.

rve stands for rotational, vibrational, electronic.

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: \(T^{nu} = \frac{1}{2} \sum _{\alpha = 1}^N m_\alpha \dot{{\varvec{R}}}^\alpha _{LF} \cdot \dot{{\varvec{R}}}^\alpha _{LF}\).

For diatomic molecules (\(N = 2\)), the number of internal nuclear coordinates is equal to \(3N-5 = 1\).

Traditionally, \( m = 0, 1, 2, \ldots \) and m = 0 corresponds to the ground state.

These transitions generally correspond to wavelengths in the ultraviolet-visible domain.

We assume that the ground state potential energy surface has at least one local minimum.

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.

See, for instance, Chap. 13 in Ref. [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.

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 \(\langle \cdots \vert \cdots \rangle _{{{\varvec{r}}}}\).

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.

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.

See Eqs. (3.203) and (3.204) in Sect. 3.6.3.

Cohen-Tannoudji C, Dupont-Roc J, Grynberg G (1997) Photons and atoms: introduction to quantum electrodynamics. Wiley Professional

Craig DP, Thirunamachandran T (1998) Molecular quantum electrodynamics. Dover Publications

Saue T (2011) Relativistic Hamiltonians for chemistry: a primer. Chem Phys Chem 12:3077CrossRef

Bunker PR, Jensen P (1998) Molecular symmetry and spectroscopy, 2nd edn. NRC Research Press, Ottawa

Bunker PR and Jensen P (2005) Fundamentals of molecular symmetry, iop, Bristol, Philadelphia

Sutcliffe BT (1999) The idea of the potential energy surface. In: Sax AF (ed.) Potential energy surfaces. Springer, Berlin, Heidelberg, p 61

Sutcliffe BT, Wooley RG (2005) Molecular strucutre calculations without clamping the nuclei. Phys Chem Chem Phys 7:3664CrossRef

Mátyus E, Hutter J, Müller-Herold U, Reicher M (2011) Extracting elements of molecular structure from all-particle wave function. J Chem Phys 135:04302CrossRef

Cederbaum LS (2004) Born-oppenheimer approximation and beyond. In: Domcke W, Yarkony DR, Köppel H (eds) Conical intersections. World Scientific Co., Singapore, pp 3

10.

Sutcliffe BT (2000) The decoupling of electronic and nuclear motions in the isolated molecule Schrödinger hamiltonian. Adv Chem Phys 114:1

11.

Zare RN (1988) Angular momentum. Wiley, New York

12.

Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Montgomery Jr JA, Vreven T, Kudin KN, Burant JC, Millam JM, Iyengar SS, Tomasi J, Barone V, Mennucci B, Cossi M, Scalmani G, Rega N, Petersson GA, Nakatsuji H, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Klene M, Li X, Knox JE, Hratchian HP, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Ayala PY, Morokuma K, Voth GA, Salvador P, Dannenberg JJ, Zakrzewski VG, Dapprich S, Daniels AD, Strain MC, Farkas O, Malick DK, Rabuck AD, Raghavachari K, Foresman JB, Ortiz JV, Cui Q, Baboul AG, Clifford S, Cioslowski J, Stefanov BB, Liu G, Liashenko A, Piskorz P, Komaromi I, Martin RL, Fox DJ, Keith T, Al-Laham MA, Peng CY, Nanayakkara A, Challacombe M, Gill PMW, Johnson B, Chen W, Wong MW, Gonzalez C, Pople JA (2003) Gaussian 03, Revision C.02. Gaussian, Inc., Pittsburgh PA

13.

Werner H-J, Knowles PJ, Knizia G, Manby FR, Schütz M, Celani P, Korona T, Lindh R, Mitrushenkov A, Rauhut G, Shamasundar KR, Adler TB, Amos RD, Bernhardsson A, Berning A, Cooper DL, Deegan MJO, Dobbyn AJ, Eckert F, Goll E, Hampel C, Hesselmann A, Hetzer G, Hrenar T, Jansen G, Köppl C, Liu Y, Lloyd AW, Mata RA, May AJ, McNicholas SJ, Meyer W, Mura ME, Nicklass A, O’Neill DP, Palmieri P, Pflüger K, Pitzer R, Reiher M, Shiozaki T, Stoll H, Stone AJ, Tarroni R, Thorsteinsson T, Wang M, Wolf A (2010) Molpro, version 2010.1, a package of ab initio programs. http://www.molpro.net

14.

Bransden BH, Joachain CJ (2000) Quantum mechanics. Prentice Hall, Upper Saddle River, New Jersey

15.

Cohen-Tannoudji C, Diu B, Laloe F (1992) Quantum mechanics. Wiley-VCH

16.

Schatz GC, Ratner MA (2002) Quantum mechanics in chemistry. Dover Publications, New York

17.

Levine IN (1975) Molecular spectroscopy. Wiley, New-York, London

18.

Szabo A, Ostlund NS (1996) Modern quantum chemistry. Dover, Mineola, New York

19.

Herzberg GH (1966) Electronic spectra of polyatomic molecules. Van Nostrand Reihnold, Toronto

20.

Irikura KK (2007) Experimental vibrational zero-point energies: diatomic molecules. J Phys Chem 36:389

21.

Herzberg G (1992) Molecular spectra and molecular structure. Krieger Pub Co

Titel: Molecular Hamiltonian Operators
verfasst von: Fabien Gatti
Benjamin Lasorne
Hans-Dieter Meyer
André Nauts
Verlag: Springer International Publishing
Buch: Applications of Quantum Dynamics in Chemistry
Print ISBN: 978-3-319-53921-8

Electronic ISBN: 978-3-319-53923-2

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-53923-2_3

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