Quantal and classical geometric phases in molecules: Born-Oppenheimer Schrodinger equation for the electronic wavefunction

Florence J. LinUniversity of Southern California, Department of Mathematics, KAP 108, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532

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Background (cont’d.)Abstract Results

This approach identifies the internal angular momentum as the source of the geometric phase in N-body molecular dynamics:

Examples of the relationship between the orbital angular momentum and the internal angular momentum:

(i) in the spectroscopists’ Hamiltonian operator

(ii) in the classical Hamiltonian

(iii) in the quantum Hamiltonian operator

(iv) in the Born-Oppenheimer Hamiltonian operator

Institute for Mathematics and Its Applications Workshop on Mathematical and Algorithmic Challenges in Electronic Structure Theory; © 2008 F. J. Lin.

This work relates the quantal and classical geometric phases by using the classical-quantum correspondence that exists between the classical Hamiltonian and the quantum Hamiltonian operator for N-body molecular dynamics. The classical geometric phase is the net overall rotation due to the internal angular momentum; the quantal geometric phase arises due to the expected value of the internal angular momentum.

Example (ii): Classical Hamiltonianin Jacobi coordinates

The spectroscopists’ Hamiltonian suggests a classical Hamiltonian

Example (iii): Quantum Hamiltonian operator in Jacobi coordinates

Neglecting any effects of non-commuting operators, the quantum Hamiltonian operator appears in the Schrodinger equation as follows:

Eckart generalized coordinatesThe total angular momentum is

The net rotation ΔΘ of a generalized Eckart frame is

The molecular connection is

(Lin, 2007).

Overall rotation in a simulation of three-helix bundle protein dynamics

At zero total angular momentum, the flexible protein molecule rotates by 42 degrees in 105 reduced time steps due to internal motions

described in terms of bond lengths, bond angles, … (Zhou, Cook, and Karplus, 2000; see Fig. 3).

Methods

Both the quantal and classical geometric phases arise due to non-zero internal/vibrational angular momentum in N-body molecular dynamics.

(i) The classical geometric phase arises due to non-zero internal angular momentum, i.e.,

(ii) The quantal geometric phase arises due to non-zero expected value of internal angular momentum operator, i.e.,

Physical approach: (1) The classical geometric phase arises as a net overall rotation (of a generalized Eckart frame) in the center-of-mass frame as a result of the conservation of total rotational angular momentum. (2) The quantal geometric phase arises as a net phase change of the electronic wavefunction as a result of employing the time-dependent Schrodinger equation.Differential geometrical approach: Each geometric phase is explicitly expressed as the holonomy of a connection (Marsden et al., 1990).

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Summary

Discussion

References

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The quantal geometric phase [1-3] in a Born-Oppenheimer (adiabatic) electronic wavefunction is a net phase change over a closed path. Effects of the quantal geometric phase have been observed in theoretical studies of the vibrational spectra of cyclic trinitrogen (N3) molecule [4, 5]. Making a classical-quantum correspondence [6] relates the quantal geometric phase to a classical one for motion over a closed path in N-body molecular dynamics. Each is described differential geometrically as the holonomy of a connection [7], physically in terms of the internal angular momentum, and with examples. The classical geometric phase [8] is a net angle of overall rotation in the center-of-mass frame. A net rotation of 20 degrees has been observed experimentally in a triatomic photodissociation and a net overall rotation of 42 degrees has been observed computationally in protein dynamics. The Hamiltonian operator in a generalized Born-Oppenheimer Schrodinger equation for the electronic wavefunction is related to a classical Hamiltonian for N-body molecular dynamics. Both the quantal and classical geometric phases arise due to non-zero internal angular momentum in N-body molecular dynamics.

Rotation of the recoil angle in a NO2 photodissociation experiment

At zero total angular momentum, the rotation of the vector R (the change in angle of the recoil velocity of the O atom) due to the rotation of the vector r of the remaining diatomic NO fragment is described by Jacobi coordinates and is

(Demyanenko,Dribinski,Reisler, et al.,1999; see Fig. 9).

An example: Jacobi coordinates The total rotational angular momentum is

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The net overall rotation ΔθR is

(Lin, 2007).

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Example (i): Spectroscopists’ quantum Hamiltonian

The spectroscopists’ quantum Hamiltonian is

[1] C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284 (1979).[2] M. V. Berry, Proc. Royal Soc. London A 392, 45 (1984).[3] B. Simon, Phys. Rev. Lett. 51, 2167 (1983).[4] D. Babikov, B. K. Kendrick, P. Zhang, and K. Morokuma, J. Chem. Phys. 122, 044315 (2005).[5] D. Babikov, V. A. Mozhayskiy, and A. I. Krylov, J. Chem. Phys. 125, 084306 (2006).[6] F. J. Lin, Quantal and classical geometric phases, 2008.[7] J. E. Marsden, R. Montgomery, and T. Ratiu, Memoirs of the American Mathematical Society, Vol. 88, No. 436, American Mathematical Society, Providence, RI, 1990. [8] F. J. Lin, Discrete and Continuous Dynamical Systems, Supplement 2007, 655 (2007). [9] A. V. Demyanenko, V. Dribinski, H. Reisler, H. Meyer, and C. X. W. Qian, J. Chem. Phys. 111, 7383 – 7396 (1999).[10] Y. Zhou, M. Cook, and M. Karplus, Biophys. J. 79, 2902 – 2908 (2000).

Example (iv): Born-Oppenheimer Hamiltonian operator

A generalized Born-Oppenheimer Schrodinger equation is

The first kinetic energy term in the Hamiltonian operator is due to the orbital angular momentum; the kinetic energies of internal/vibrational angular momentum have been neglected here.

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Quantal and classical applications

The classical/quantum Hamiltonian is the sum of •overall rotational kinetic energy •internal/vibrational kinetic energies•potential energyWhen the orbital angular momentum in the classical/quantum Hamiltonian vanishes, the Hamiltonian depends only on the internal coordinates and their conjugate momenta.

Results (cont’d.)

Born-Oppenheimer Schrodinger equation

The Schrodinger equation and initial eigenstate satisfy

The wavefunction evolves according to the Hamiltonian and satisfies

The net phase ΔΘ of a Born-Oppenheimer (adiabatic) wavefunction is

The quantal connection is

(Lin, 2008).

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The classical geometric phase in molecules (Lin, 2007) is reviewed. A classical-quantum correspondence describes the quantal geometric phase as a consequence of the classical geometric phase (Lin, 2008).

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