(132h) Hartree Fock Self-Consistent Calculations: Global Optimization of Electronic Structure

Authors: 
Zorn, Jr., K., Carnegie Mellon University
Sahinidis, N., Carnegie Mellon University


Hartree-Fock (HF) theory is a set of assumptions and approximations in ab-initio quantum chemistry. Under HF, independent electronic motion results from mean fields involving electron-nucleus attraction, electron-electron repulsion, and electron-electron exchange energies [1]. The variational theorem states that approximations to the true electronic wavefunction yield ground-state energies that are higher than the energy of the true system [2]. A set of coupled, nonlinear differential equations results and the most accurate wavefunction is determined as that which produces the smallest ground-state energy.

Hartree-Fock theory and the linear combination of atomic orbitals (LCAO) approximation yield a finite and explicit definition of total energy. For these HF systems, the resulting energy equation is a polynomial expression in terms of LCAO coefficients [3]. Hartree-Fock systems are typically solved by self-consistent field (SCF) iteration, but combining the polynomial energy equation with constraints ensuring self-consistency and basis set orthogonality produces a continuous nonlinear minimization problem.

One of the main challenges associated with the solution of the above models is the multimodal nature caused by the inclusion of highly nonlinear and nonconvex equations. Tight convex relaxations are essential when solving these difficult problems to global optimality with a branch-and-bound algorithm [4]. It is now understood that the successful application of convexification techniques often requires unconventional mathematical reformulations of the problem constraints. We combine reformulation linearization techniques (RLT) [5] and advanced convex envelope construction techniques [6] to produce tight subproblem formulations. The quantum chemical application is used to gain insights to the problem of identifying strong relaxations with the aim of producing tight, lower-dimensional problem formulations. Computational results involving small HF systems are presented.

References:

[1] I. Levine. Quantum Chemistry. Pearson Education Incorporated. Sixth Edition. 2009.

[2] C. Cramer. Computational Chemistry: Theories and Models. John Wiley and Sons Limited. Second Edition. 2004.

[3] L. Liberti, C. Lavor, N. Maculan, and M.A.C Nascimento. ?Reformulation in mathematical programming: An application to quantum chemistry.? Discrete Applied Mathematics, 157(6) (2009): 1309-1318.

[4] M. Tawarmalani and N. V. Sahinidis. "Global optimization of mixed-integer nonlinear programs: A theoretical and computational study." Mathematical Programming, 99(3) (2004): 563-591.

[5] H.D. Sherali and C.H. Tuncbilek. ?New reformulation linearization/convexification relaxations for univariate and multivariate Polynomial programming problems.? Operations Research Letters, 21 (1997): 1-9

[6] M. Tawarmalani and N. V. Sahinidis. Convex extensions and convex envelopes of l.s.c. functions. Mathematical Programming, 93 (2002): 247?263.