1. Quantal and classical geometric phases in molecules: Born-Oppenheimer Schrodinger equation for the electronic wavefunction Florence J. Lin University of Southern California, Department of Mathematics, KAP 108, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532 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 Hamiltonian in 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 coordinates The 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 10 5 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). Background Objective Summary Discussion References 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 (N 3 ) 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 NO 2 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 with The net overall rotation Δθ R is (Lin, 2007). 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. Quantal and classical applications The classical/quantum Hamiltonian is the sum of overall rotational kinetic energy internal/vibrational kinetic energies potential energy When 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). 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).