Batista_AIMD_Sim - 2006 Summer School on Computational...

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Unformatted text preview: 2006 Summer School on Computational Materials Science Lecture Notes: Ab Initio Molecular Dynamics Simulation Methods in Chemistry Victor S. Batista * Yale University, Department of Chemistry, P.O.Box 208107, New Haven, Connecticut 06520-8107, U.S.A. I Introduction These lectures will introduce computational methods that provide quantum mechanical descriptions of the dynamical and equilibrium properties of polyatomic systems. 1–7 According to the fifth postulate of quantum mechanics, the description of dynamics requires solving the time-dependent Schr¨odinger equation i ∂ Ψ t ( x ) ∂t = ˆ H Ψ t ( x ) , (1) subject to a given initial condition, Ψ ( x ) . To keep the notation as simple as possible, all expressions are written in atomic units, so ~ = 1 . Here, ˆ H = ˆ p 2 / (2 m ) + V (ˆ x ) is the Hamiltonian operator, ˆ p =- i ∇ is the momentum operator and V (ˆ x ) is the potential energy operator. A formal solution of Eq. (1) can be obtained by integration, as follows: Ψ t ( x ) = Z dx h x | e- i ˆ Ht | x ih x | Ψ i , (2) where the Kernel h x | e- i ˆ Ht | x i is the quantum propagator. As an example, consider a diatomic molecule vibrating near its equilibrium position ¯ x where the potential is Harmonic, V (ˆ x ) = 1 2 mω 2 (ˆ x- ¯ x ) 2 . (3) The description of the time-dependent bond-length x ( t ) is given by the expectation value x ( t ) = h Ψ t | ˆ x | Ψ t i , (4) where Ψ t is defined according to Eq. (2) with the particular Kernel, h x | e- i ˆ Ht | x i = r mω 2 π sinh ( itω ) exp- mω 2 sinh ( ωit ) [( x 2 + x 2 ) cosh ( ωit )- 2 xx ] . (5) This standard formulation of quantum mechanics relies upon the tools of calculus ( e.g. , derivatives, integrals, etc.) and involves equations and operations with infinitesimal quantities as well as states in Hilbert- space (the infinite dimensional space of functions L 2 ). These equations, however, seldom can be solved analytically as shown in the example above. Therefore, computational solutions are necessary. However, computers can not handle infinite spaces since they have only limited memory. In fact, all they can do is to store and manipulate discrete arrays of numbers. Therefore, the question is: how can we represent continuum states and operators in the space of memory of digital computers? * E-mail: [email protected] 1 II Grid-Based Representations In order to introduce the concept of a grid-representation, we consider the state, Ψ ( x ) = α π 1 / 4 e- α 2 ( x- x ) 2 + ip ( x- x ) , (6) which can be expanded in the infinite basis set of delta functions δ ( x- x ) as follows, Ψ ( x ) = Z dx c ( x ) δ ( x- x ) , (7) where c ( x ) ≡ h x | Ψ i = Ψ ( x ) ....
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Batista_AIMD_Sim - 2006 Summer School on Computational...

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