STOCHASTIC AND DETERMINISTIC MOLECULAR DYNAMICS DERIVED FROM THE TIME-INDEPENDENT SCHRÖDINGER EQUATION

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1 STOCHASTIC AND DETERMINISTIC MOLECULAR DYNAMICS DERIVED FROM THE TIME-INDEPENDENT SCHRÖDINGER EQUATION ANDERS SZEPESSY Abstract. Born-Oppenheimer, Smoluchowski, Langevin, Ehrenfest and surface-hopping dynamics are shown to be accurate approximations of time-independent Schrödinger observables for a molecular system, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on characteristics for the Schrödinger equation, bypasses the usual separation of nuclei and electron wave functions and gives a different perspective on initial and boundary conditions, caustics and irreversibility, the Born-Oppenheimer approximation, hopping, computation of observables and stochastic electron equilibrium states in molecular dynamics modeling. Contents 1. The Schrödinger and Ehrenfest models 2 2. Ehrenfest dynamics derived from the time-independent Schrödinger equation Exact Schrödinger dynamics Irreversibility in Hamilton-Jacobi equations generated by shock waves Approximate Ehrenfest dynamics and densities Comparison of two alternative Ehrenfest formulations Equations for the density Construction of the solution operator The time-dependent Schrödinger equation Surface-hopping and multiple states Surface-hopping and Ehrenfest dynamics for multiple states Characteristics hopping in the Ehrenfest dynamics Computation of observables 2 6. Approximation error derived from a Hamilton-Jacobi equation The Ehrenfest approximation error The Born-Oppenheimer approximation Stochastic Langevin and Smoluchowski molecular dynamics approximation Construction of the solution operator 32 2 Mathematics Subect Classification. Primary: 81Q2; Secondary: 82C1. Key words and phrases. Ehrenfest dynamics, quantum classical molecular dynamics, surface-hopping, foundations of quantum mechanics, WKB expansion, Born-Oppenheimer approximation, Langevin equation, Smoluchowski equation, Schrödinger equation, shock wave. 1

2 2 ANDERS SZEPESSY 7.1. Spectral decomposition Derivatives of the wave function Discrete spectrum The Born-Oppenheimer approximation 37 References The Schrödinger and Ehrenfest models The time-independent Schrödinger equation (1.1) H(x, X)Φ(x, X) = EΦ(x, X), models nuclei-electron systems and is obtained from minimization of the energy in the solution space of wave functions, cf. [39, 4, 41, 1]. It is an eigenvalue problem for the energy E R of the system in the solution space, described by wave functions, Φ : R 3J R 3N C, depending on electron coordinates x = (x 1,..., x J ) R 3J, nuclei coordinates X = (X 1,..., X N ) R 3N, and a Hamiltonian operator H(x, X) H(x, X) = V (x, X) 1 2M N X n. The nuclei masses M are assumed to be large and the interaction potential V, independent of M, is in the canonical setting (neglecting relativistic and magnetic effects), (1.2) V (x, X) = 1 2 N J n=1 =1 J x + =1 1 k< J Z n x X n + n=1 1 n<m N 1 x k x Z n Z m X n X m, composed of the kinetic energy of the electrons, the electron-electron repulsion, the electronnuclei attraction, and the repulsion of nuclei (with charge Z n ), in the Hartree atomic units where the electron mass, electron charge, reduced Planck constant, and the Coulomb force constant (4πɛ ) 1 all are one. The mass of the nuclei, which are much greater than one (electron mass), are the diagonal elements in the diagonal matrix M. An essential feature of the partial differential equation (1.1) is the high computational complexity to determine the solution, in an antisymmetric/symmetric subset of the Sobolev space H 1 (R 3(J+N) ). Wave functions depend also on discrete spin states (1.3) Φ(x 1, σ 1,..., x J, σ J, X 1, Σ 1,..., X N, Σ N ), which effect the solutions space: each electron spin σ can take two different values and each nucleus can be in a finite set of spin states Σ n ; the Pauli exclusion principle restricts the

3 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 3 solutions space to wave functions satisfying the antisymmetry/symmetry Φ(..., x, σ,..., x k, σ k,...) = Φ(..., x k, σ k,..., x, σ,...) for any 1, k J and for any pair of nuclei n and m, with A nucleons and the same number of protons and neutrons, Φ(..., X m, Σ m,..., X n, σ n,...) = ( 1) A Φ(..., X n, Σ n,..., X m, Σ m,...) cf. [1]. We simplify the notation by writing Φ(x, X) instead of the more complete (1.3), since the Hamiltonian H does not depend on the spin of each particle. An attractive property of the Schrödinger equation (1.1) is the precise definition of the Hamiltonian and the solutions space, without unknown parameters, and its good agreement with experimental results. The agreement with measurements can be further improved by including relativistic and magnetic effects, cf. [1]. In contrast to the Schrödinger equation, a molecular dynamics model of nuclei X : [, T ] R 3N, with a given potential V : R 3N R, can be computationally studied also for large N by solving the ordinary differential equation (1.4) MẌτ = X V (X τ ). This computational and conceptual simplification motivates the study to determine the potential and its implied accuracy by a derivation of molecular dynamics from the Schrödinger equation, as started already in the 192 s with the seminal Born-Oppenheimer approximation [7]. The purpose here is to contribute to the current understanding of such derivations, by showing improved convergence rates under new assumptions. A useful sub step to derive molecular dynamics from the Schrödinger equation is Ehrenfest dynamics, for classical ab initio motion of the nuclei coupled to Schrödinger dynamics for the electrons, MẌn τ = φ τ(, X τ ) X nv (, X τ ) φ τ (, X τ ) dx (1.5) R 3J i φ τ = V (, X τ )φ τ. The Ehrenfest dynamics (1.5) has been derived from the time-dependent Schrödinger equation through the self consistent field equations, see [8, 36, 42]. Equation (1.5) can be used for ab initio computation of molecular dynamics, cf. [36, 33]. A next step is the Born-Oppenheimer approximation, where X τ solves the classical ab initio molecular dynamics (1.4) with the potential V : R 3N R determined as an eigenvalue of the electron Hamiltonian V (, X) for a given nuclei position X, that is V (, X)φ (X) = V (X)φ (X), for instance with the electron ground state φ (X). The Born-Oppenheimer approximation has been derived from the time-dependent Schrödinger equation in [24, 38]. The model (1.4) simulates dynamics at constant energy M Ẋ 2 /2+V (X), constant number of particles N and constant volume, i.e. the microcanonical ensemble. The alternative to simulate with constant number of particles, constant volume and constant temperature T,

4 4 ANDERS SZEPESSY i.e. the canonical ensemble, is possible for instance with the stochastic Langevin dynamics (1.6) dx τ = v τ dτ Mdv τ = X V (X t )dτ Kv t dτ + (2T K) 1/2 dw τ, where W τ is the standard Brownian process (at time τ) in R 3N with independent components and K is a positive friction parameter. When the observable only depends on the nuclei positions, i.e. not on the nuclei velocities or the correlation of positions at different times, the Smoluchowski dynamics (1.7) dx τ = X V (X τ ) + (2T ) 1/2 dw τ is a simplified alternative to Langevin dynamics, cf. [9]. This paper derives the Ehrenfest dynamics (1.5) and the Born-Oppenheimer approximation from the time-independent Schrödinger equation (1.1). The time-independent Schrödinger equation has convincing agreement with experimental results, as the basis for computational chemistry and solid state physics. The time-dependent Schrödinger equation on the other hand is less well established from experiments and that motivates also to present a derivation of molecular dynamics from the time-independent Schrödinger equation, although the main point here is to show improved convergence rates under simple assumptions. The main idea in this paper, inspired by [37, 5, 6], is to introduce the time-dependence from the classical characteristics in the Hamilton-Jacobi equation obtained by writing the time-independent eigenfunction (1.1) in WKB-form. The work [37, 5, 6] derives the timedependent Schrödinger dynamics of an x system, i Ψ = H 1 Ψ, from the time-independent Schrödinger equation (with the Hamiltonian H 1 (x) + δh(x, X)) by a classical limit for the environment variable X, as the coupling parameter δ vanishes and the mass M tends to infinity; in particular [37, 5, 6] show that the time derivative enters through the coupling of Ψ with the classical velocity. Here we refine the use of characteristics to study classical ab initio molecular dynamics, where the coupling does not vanish, and we establish error estimates for Born-Oppenheimer, Ehrenfest, surface-hopping and stochastic approximations of Schrödinger observables. The small scale, introduced by the perturbation (2M) 1 n X n of the potential V, is identified in a standard WKB eikonal equation, while its transport equation is analyzed as a timedependent Schrödinger equation along characteristics, instead of as a usual series expansion [26]. The analysis based on the characteristics for the time-independent Schrödinger equation, bypasses the usual separation of nuclei and electron wave functions in the time-dependent self consistent field equations [8, 36, 42]. The characteristic particle paths that may or may not return to the inflow domain, gives a different perspective on initial and boundary conditions, caustics and irreversibility, the Born-Oppenheimer approximation, hopping, and computation of observables. Section 3 shows that the Ehrenfest and surface-hopping dynamics are the same when derived from the time-independent and time-dependent Schrödinger equations. Theorems 6.1 and 6.3 present conditions for approximating observables from the Schrödinger equation by observables from the Ehrenfest dynamics and the Born-Oppenheimer dynamics

5 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 5 with error O(M 1 ) respectively O(M 1/2 ), using that these approximate solutions generate approximate eigenstates to the Schrödinger equation. The derivation does not assume that the nuclei are supported on small domains; in contrast, derivations based on the timedependent self consistent field equations require nuclei to be supported on small domains. The reason that small support is not needed here comes from the combination of the characteristics and sampling from an equilibrium density, that is, for large M the nuclei paths behave classically although they may not be supported on small domains. A large nuclei system, N M, is used in Remark 6.2 to motivate an assumption on perturbations of the solutions to the Schrödinger equation (1.1). The derived approximation rates improve the previous O(M 1/2 ) rate for Ehrenfest [8] and O(M 1/4 ) rate for the zero order Born-Oppeneheimer approximation [24]. Section 6.3 applies the Ehrenfest approximation result to derive the Langevin and Smoluchowski dynamics from the Ehrenfest dynamics, when the electron state is stochastic perturbed from its ground state and the observable depends only on the nuclei positions but not their correlation at different time. The derivation uses a classical equilibrium Gibbs- Boltzmann distribution for the electron states and an assumption of a spectral gap, showing in Theorem 6.6 that observables of Langevin and Smoluchowski dynamics accurately approximate such Schrödinger observables. The main idea in the theorem is the non-standard view of a classical Gibbs-Boltzmann equilibrium distribution of electrons states, motivated by nuclei acting as heat bath for the electrons in the Ehrenfest Hamiltonian system. It is my hope that the ideas in this paper stimulates more research on the conditions for molecular dynamics approximation. In particular it would be desirable to have more explicit conditions for the L 2 (dxdx)-bound on the X-Laplacian of the Ehrenfest electron wave function in (6.3), used in the approximation results and motivated in Remark 6.2, and further investigate the irreversibility of molecular dynamics caused by colliding characteristic paths, presented in Section Ehrenfest dynamics derived from the time-independent Schrödinger equation 2.1. Exact Schrödinger dynamics. Assume for simplicity that all nuclei have the same mass. 1 The singular perturbation (2M) 1 n X n of the potential V introduces an additional small scale M 1/2 of high frequency oscillations, as shown by a WKB-expansion, see [26, 12]. We will construct solutions to (1.1) in such WKB-form (2.1) Φ(x, X) = ψ n (x, X)e im 1/2 θ n(x), 1 If this is not the case, change to new coordinates M 1/2 1 to the form we want V (x, M 1/2 1 M 1/2 X) 1 2M 1 N n=1 Xn. X k = M 1/2 k X k, which transform the Hamiltonian

6 6 ANDERS SZEPESSY where the wave function ψ n is complex valued, the phase θ n is real valued and the factor M 1/2 is introduced to have well defined limits of ψ n and θ n as M. The standard WKBconstruction [26] is based on a series expansion in powers of M 1/2, we introduce instead a timedependent Schrödinger transport equation. In the next section we use a linear combination of such eigensolutions - therefore the index n is introduced. The Schrödinger equation (1.1) implies that (2.2) = (H E)ψ n e im 1/2 θ n(x) = (( Xθ n 2 + V E)ψ n 2 1 X ψ n i ( 2M M 1/2 X ψ n X θ n + 1 ) 2 ψ n X X θ n) e im 1/2 θ n(x). Introduce the complex-valued scalar product v w := v(x, ) w(x, )dx v w R 3J on L 2 (R 3J ) and the notation X Y for the standard scalar product on R 3N. To find an equation for θ n, multiply (2.2) by ψ n and integrate over R 3J ; similarly take the complex conugate of (2.2), multiply by ψ n and integrate over R 3J ; and finally add the two expressions to obtain (2.3) = 2 ( X θ n ( ψ n ( 2M + i ( M 1/2 E ) ψ n ψ n + ψ n V ψ n + V ψ n ψ }{{ n } =2ψ n V ψ n X ψ n ) + ( ) X ψ n ) ψ n ψ n ( X ψ n X θ n ) ( X ψ n X θ n ) ψ }{{ n ( ) } =2iI ψ n ( X ψ n X θ) + i ( ) (ψ 2M 1/2 n ψ n ψ n ψ n ) }{{} X X θ n. = The purpose of the phase function θ n is to generate an accurate approximation in the limit as M : therefore we define θ n by the formal limit of (2.3) as M, which is the Hamilton-Jacobi equation (also called the eikonal equation) (2.4) where the function V n : R 3N R is X θ n 2 2 = E V n, V n := ψ n V ψ n ψ n ψ n. )

7 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 7 Define also the density function ρ := ψ n ψ n. For the energy E chosen larger than the potential energy, that is such that E V n, the method of characteristics, cf. [16], (2.5) dx t = X θ n (X t ) =: p t, dt dp t dt = XV n (X t ), dz t dt = p t 2 = 2 ( E V n (X t ) ), yields a solution (X, p, z) : [, T ] U R 3N R to the eikonal equation (2.4) locally in a neighborhood U R 3N, for regular compatible data (X, p, z ) given on a 3N 1 dimensional inflow-domain I U; here z t := θ n (X t ). The work [26] proves the existence of a C solution θ n to the eikonal equation (2.4) in a neighborhood of a global minimum point of a given C non negative potential E V n : R 3N [, ); this handles the type of caustic (i.e. colliding characteristics) where X θ n vanishes. The phase function θ n : U R becomes globally defined in U R 3N for instance when there is a unique characteristic path X t going through each point in U. Section 2.2 presents conditions for the case when characteristic paths collide, that lead to irreversible molecular dynamics. Definition (2.4) implies that ψ n solves the so called transport equation (2.6) 1 2M X ψ n + (V V n )ψ n = i M 1/2 ( X ψ n X θ n X X θ nψ n ). Time enters into the Schrödinger equation through the characteristics and the chain rule X ψ n X θ n = X ψ n dx t dt = dψ n(x, X t ), dt cf. [37, 5, 6]. Chose a coordinate system with the X 1 1-direction parallel to the momentum p = (p 1 1,,..., ) to evaluate the divergence (2.7) X X θ n = div p = p1 1 X 1 1 = p 1 1 t X 1 1 t = ṗ1 1 p 1 1 = d dt p 2 2 p 2 = d log p dt which leads to the integrating factor G t := p t with the time derivative d dt Ġ = G p 2 p = G div p; 2 here our derivation differs from [37, 5, 6] which approximates the last term in (2.6), X X θ nψ n, by zero in their case of vanishing coupling between the quantum system and the environment; here the coupling between the nuclei and electrons does not vanish. The right hand side

8 8 ANDERS SZEPESSY in (2.6) becomes the time derivative im 1/2 G 1 d(gψ n )/dt and we have derived the timedependent Schrödinger equation, for the variable ψ n := Gψ n, (2.8) The density can be written = (H E)Φ ( ( i = ψn + (V V M 1/2 n ) ψ n 1 G X ( 2M ψ n G 1 ) ) G 1 + ( X θ n 2 + V n E ) ψ n )e im 1/2 θ n(x). } 2 {{} = ρ = ψ n 2 dx = R 3J ψ n 2 dx/g 2 R 3J and therefore the second equation in (2.5) yields the nuclei dynamics Ẍ = X ψn V ψ n ψ n ψ n. In conclusion, we have derived the exact Schrödinger dynamics along the characteristics i ψn = (V V M 1/2 n ) ψ n 1 G X ( 2M ψ/g), (2.9) ψn V n ψn Ẍ = X ψ n ψ. n The derivatives X G and XX G can be determined from (p, X p, XX p), which satisfy the following characteristic equations obtained from X-differentiation of (2.4) d [ dt X rpk = p X X rpk = ] p X r X kp (2.1) = X rp X kp X r X kv n, d [ dt X r X qpk = p X X r X qpk = ] p X r X k X qp = X rp X k X qp X r X qp X kp X r X k X qv n Irreversibility in Hamilton-Jacobi equations generated by shock waves. We will here show that colliding characteristic paths lead to irreversible Ehrenfest dynamics, although it is a Hamiltonian system. We will do that by requiring stability of the Hamilton- Jacobi and Schrödinger system (2.9) for a case with colliding characteristic paths.

9 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 9 We do not need a unique solution of the Hamilton-Jacobi equation, since we study an eigenvalue problem with multiple solutions, but we want to sort out stable solutions: if the data for θ n and ψ n is slightly perturbed in the maximum norm, there should be a solution θ n and ψ n which is close to the non perturbed solution in maximum norm. A stable generalized solution to the Hamilton-Jacobi equation with colliding paths can be irreversible in the sense that the solution is only stable when the initial data is given and not the final data, as explained in the example below. At a point X of two colliding characteristic paths, (X, p, z ) and (X, p +, z + ), we need first that (2.11) the constructed phase θ n is continuous, i.e. z = z +. Typically this condition yields a co-dimension one surface Γ in R 3N (called a shock wave) where the characteristic paths collide; at the other points in R 3N \ Γ the characteristic paths do not collide. Secondly, to have a stable solution ψ n = p 1/2 ψn, the right hand side in the Schrödinger equation (2.9) i (2.12) ψn = (V V M 1/2 n ) ψ n p 1/2 X ( p 1/2 ψn ), 2M has to be well behaved in the neighborhood of Γ, which excludes umps in p ; to have such umps in p means that the Hamiltonian is continuous, i.e. (2.13) p + 2 = p 2. At the collision surface Γ, the Schrödinger equation (2.12) then has a right hand side where the function p satisfies p + 1/2 = p 1/2 X p + 1/2 = X p 1/2 X p + 1/2 = X p 1/2, so that although p has a ump, the function p is smooth in a neighborhood of Γ. The solution ψ n is therefore bounded also in a neighborhood of Γ. The requirement to satisfy the ump condition (2.13) is also one of the necessary conditions for unique so called viscosity solutions of Hamilton-Jacobi equations, see [16], but (2.13) and (2.11) are not enough, see the example below. Our third condition is to require that (2.14) the solution θ n is maximum norm stable towards perturbations of the initial data. To more precisely define irreversibility we will use an example. We seek stable solutions to the Hamilton-Jacobi equation F ( X θ(x), X ) =, using its Hamiltonian system Ẋ t = p t ṗ t = X V (X t ),

10 1 ANDERS SZEPESSY which has formally reversible solutions: for T >, the initial data p (X ) determines (X T, p T ), which is the same as (X T, p T ) determined from the final data p (X ). Consider the Hamilton-Jacobi equation p p = in the domain U = R [ 1, ], in R 2, with the associated Hamiltonian system Ẋ t = p t ṗ t =. This Hamilton-Jacobi equation has the stable initial-value solution { 1 X1 < p 1 = (2.15) 1 X 1 >, p 2 = 1 and the colliding characteristic paths satisfy the ump condition p = p + at Γ = {} [ 1, ]. The corresponding reversed final-value solution { 1 X1 < p 1 = 1 X 1 >, p 2 = 1, which also satisfies p = p + at Γ, is unstable with respect to perturbations in the final value data: the regularized perturbed boundary data 1 X 1 < δ p 1 = δ 1 X 1 δ < X 1 < δ 1 X 1 > δ, p 2 = 1, as final value at the the inflow domain X 2 =, leads to the different stable final-value solution p(x t ) = p(x ) = ( p 1 (y, ), 2 p 1 (y, ) 2) (2.16) X t = X + t p(x ) t <, with no ump in p, so in particular this p is not the negative of (2.15) as required for reversibility. The two stable solutions (2.15) and (2.16) are examples of a shock wave and a rarefaction wave, respectively, cf. [16]. Requiring such stability, solutions of the Hamilton-Jacobi equation and solutions to the corresponding Hamiltonian system are not reversible with respect to giving initial and final condition. In this sense, the molecular dynamics model (2.9) is time-irreversible. The solution of the Hamilton-Jacobi equation, satisfying the three conditions (2.11), (2.13) and (2.14), is in fact the unique viscosity solution of F ( X θ(x), X ) := X θ(x) 2 /2 + V (X) E = for X U, θ(x) = g(x) or F ( X θ(x), X ) = for X U

11 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 11 for given functions V : R 3N R and g : R 3N R; the stability holds with respect to perturbations in the boundary data g and the potential V, in the maximum norm, and with respect to the domain U, see [2], [3] and [16]. The stable solution to the final-value problem, (X T, p T ), with respect to perturbations in the final data p (X ), can therefore be characterized by the optimal control problem θ f (Y ) = inf X T = Y, T > ( T Ẋt 2 2 ) V (X) + E dt g(x ), where t = is the time a path X t, starting at time t = T in X T = Y U, reaches the boundary U; this viscosity solution can also be described as the vanishing viscosity solution, i.e. θ f = lim ɛ + θ ɛ where F ( X θ ɛ (X), X ) + ɛ X θ ɛ (X) = X U θ ɛ (X) = g(x) X U, cf. [2], [3] and [16]. Analogously, the stable solution to the initial value problem (X T, p T ), with respect to perturbations in the initial data p (X ), is the corresponding viscosity solution θ i to F ( X θ i (X), X ) =, which can be characterized by the optimal control problem θ i (Y ) = inf X T = Y, T > ( T or by the vanishing viscosity limit θ i = lim ɛ + θ ɛ where F ( X θ ɛ (X), X ) ɛ Ẋt 2 ) V (X) + E dt + g(x ), 2 X θ ɛ (X) = θ ɛ (X) = g(x) X U X U. The example showed that θ f (X) θ i (X) and hence stable solutions can be time-irreversible by only requiring stability with respect to perturbations in the initial data; the obtained viscosity solution of the Hamilton-Jacobi equation is then a consequence of this stability requirement. We end the section by two examples of more complicated collisions. If characteristic paths collide, they may form closed curves Γ containing crossing points and to avoid multiple values of θ n we need that the corresponding integral satisfies b p p dt = p dx =, a Γ which follows by Stokes theorem since p = X θ n locally. When three characteristic paths, X +, X and X collide into a point it yields a collision of three shock waves. The codimension one surface Γ in R 3N consists now of three parts, which intersect where the three paths have the same value of θ n. The three parts are generated, as in above, by the θ n -value attained by two paths X + &X, X + &X, and X &X, respectively. This can be generalized to up to 3N + 1 colliding paths.

12 12 ANDERS SZEPESSY 2.3. Approximate Ehrenfest dynamics and densities. We define the approximating Ehrenfest dynamics by in (2.2) neglecting the kinetic nuclei term (2.17) ( X ˆθn 2 = 2 ) + V E ˇψn i ( M ˇψ 1/2 X n ˆθ X n ˇψ n ˆθ X X n ), and seek, as in (2.4), the approximate phase ˆθ n as the solution to eikonal equation (2.18) where Introduce its characteristics to rewrite (2.17), as in (2.8), X ˆθn 2 2 = E ˆV n, ˆV n := ˇψ n V ˇψ n ˇψ n ˇψ n. d ˆX t = X ˆθn (X t ) =: ˆp t, dt dˆp t dt = ˆV X n ( ˆX t ), dẑ t dt = ˆp t 2 = ˆV n ( ˆX t ), (2.19) = ( i ˆψn (V M ˆV 1/2 n ) ˆψ )Ĝ 1 n + ( X ˆθn 2 + V n E ) } 2 {{} = ˆX = X( ˆψ n ˆV n ˆψn ) ˆψ n ˆψ n, ˇψ n, for ˆψ n := Ĝ ˇψ n approximating ψ n and Ĝ := ˆp t as in (2.7). Using that ˆψ n ˆψ n is conserved (i.e. time-independent) in the Ehrenfest dynamics, we can normalize to ˆψ n ˆψ n = 1. Note that in the exact dynamics, the function ψ n ψ n is not conserved, due to the L 2 (R 3J ) non-hermitian source term 1 2M G X ( ψ n /G) in (2.9). We have V n = ψ n V ψ n ψ n ψ n = ψ n V ψ n ψ n ψ n and ˆV n = ˆψ n V ˆψ n ˆψ n ˆψ n = ˆψ n V ˆψ n.

13 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION Comparison of two alternative Ehrenfest formulations. The Ehrenfest dynamics (2.19) and (1.5) differ in (1) the time scales t respectively M 1/2 t = τ; (2) the potentials V V n and V in the equations for ˆψ n and ˆφ n, respectively; and (3) the forces X ( ˆψ n V ˆψ n ) respectively ˆφ n X V ˆφ n in the momentum equation. If ˆψ n solves (2.19), the change of variables ˆφ n = ˆψ n e im 1/2 R t ˆψ n V ˆψ n(x s)ds and the property ˆψ n A ˆψ n = ˆφ n A ˆφ n, which holds for any observable A independent of the nuclei momentum i X (in particular A = V n ), imply that ˆφ n solves (2.2) i M 1/2 ˆφn = V ˆφ n, ˆX = X( ˆφ n ˆV n ˆφn ) ˆφ n ˆφ n. There has been a discussion in the literature [8, 42] whether the forces should be computed as above in (2.2) or as in (1.5) by (2.21) MẌn τ = φ τ(, X τ ) X nv (, X τ ) φ τ (, X τ ) dx R 3J i φ τ = V (, X τ )φ τ. The work [43] points out that the two forces are the same in the direction of p(x τ ), that is ( ) p(x τ ) φ τ X nv (, X τ ) φ τ dx X n φ τ(, X τ ) V (, X τ ) φ τ (, X τ ) dx =. R 3J R 3J Equation (6.5) show that both formulations (2.2) and (2.21, 1.5) yield accurate approximations of the Schrödinger observables, although they are not the same. The reason that both approximations are accurate is that the two different characteristic systems solve the same Hamilton-Jacobi equation, as explained below and in Section 6. The formulation (1.5) and (2.21) has the advantage to be a closed Hamiltonian system: the variable (X, ϕ r ; p, ϕ i ), with the definition ϕ := ϕ r + iϕ i := 2 1/2 M 1/4 φ = 2 1/2 M 1/4 (φ r + iφ i ), ϕr θn =: ϕ i,

14 14 ANDERS SZEPESSY and the Ansatz θ = ˆθ imply that the Hamilton-Jacobi equation (2.18) becomes H E := 1 2 X θ n X θn + φ r V (X)φ r + φ i V (X)φ i = 1 2 X θ n X θn M 1/2 ϕ r V (X)ϕ r M 1/2 ϕ i V (X)ϕ i = 1 2 X θ n X θn M 1/2 ϕ r V (X)ϕ r M 1/2 ϕr θn V (X) ϕr θn = E, where the derivative ϕr θ(x, ϕr ) = ϕ i, of the functional θ : R 3N L 2 (dx) R, is the Gateaux derivative in L 2 (dx). Its characteristics form the Hamiltonian system Ẋ t = p t ṗ t = M 1/2 2 ϕ(t) XV (X)ϕ(t) ϕ r (t) = M 1/2 V (X)ϕ i (t) ϕ i (t) = M 1/2 V (X)ϕ r (t), which is the same as the Hamiltonian system (2.21) (2.22) Ẋ t = p t ṗ t = φ t X V (X t )φ t i M φ 1/2 t = V (X t )φ t. The Hamiltonian system yields a modified equation for the phase θ = X ˆθn Ẋ + ϕ r ˆθn ϕ r = p p + 2φ r V φ r, since θ = θ(x, ϕ r ) is a function of both X and ϕ r in this formulation, and we see that θ is indeed a function of X, independent of x, so that the identification ˆθ = θ is possible. The important property of this Hamiltonian dynamics is that (X, p, ˆψ n ), with ˆψ n = φe im 1/2 R t φ V φ(s)ds, solves both the Hamilton-Jacobi equation (2.18) and (2.17), written as the time-dependent Schrödinger equation (2.19). The alternative (2.2) does not form a closed system, in the sense that the required function X ψ n (X t ) is not explicitly determined along the characteristics, but can be obtained from values of ψ n, in a neighborhood of X t, by differentiation.

15 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION Equations for the density. We note that ( ρn ) 1/2 ψ n = ψ n ψ ψn, n ˇψ n = ( ˆρn ˆψ n ˆψ n ) 1/2 ˆψn, shows that the densities ρ n = ψ n ψ n and ˆρ n := ˇψ n ˇψ n, in addition to (X, p, ψ n ) and ( ˆX, ˆp, ˆψ n ), are needed to determine the wave functions ψ n and ˇψ n. Equation (2.17) and the proections in (2.3) subtracted (instead of added) imply that the approximate density ˆρ n satisfies = ( ˇψ ˇψ X n n + ˇψ n ˇψ X n )dx ˆθ X n + ˇψ n ˇψn dx ˆθ X X n R 3J R 3J = X (ˆρ X ˆθ n ) and consequently, the density can be determined along a characteristic using (2.7) ˆρ( ˆX t ) = ˆX ˆρ( ˆX t ) ˆX = ˆX ˆρ( ˆX t ) ˆX ˆθ n = ˆρ( ˆX t ) ˆX ˆX ˆθ n with the solution = ˆρ( ˆX t ) div ˆp = ˆρ( ˆX t ) d dt log ˆp t ˆρ( ˆX t ) = C ˆp t, where C is a positive constant for each characteristic. Change to coordinates ˆX 1 1 R parallel and ˆX R 3N 1 orthogonal to the characteristic direction ˆX to obtain the factorization (2.23) ˆρ( ˆX)d ˆX = ˆρ( ˆX ) using and the normalization d ˆX ˆX1 1 (T ) d ˆX 1 1 ˆp 1 1 d ˆX 1 1 ˆp 1 1 = d ˆX 1 1 d ˆX 1 1/dt = dt I R 3(J+N) ˆρ( ˆX )d ˆX = 1. d ˆX 1 1 ˆp 1 1 = ˆρ( ˆX )d ˆX dt T,

16 16 ANDERS SZEPESSY Similarly, the exact Schrödinger density satisfies ρ(x t ) = X ρ(x t )Ẋ (2.24) = ρ(x t ) d dt log p t + M 1/2 I(ψ n X ψ n ), where Iw denotes the imaginary part of w. Different characteristic paths ˆX n (t) R 3N have different densities. The density is therefore important to weight the different paths, in particular because the density in the time-independent case does not follow characteristics as in the time-dependent setting, which is explained in Section Construction of the solution operator. The WKB-form (2.1) is meaningful when ψ n does not include the small scale, that is, we need to verify that X ψ n is bounded independent of M, which we do by using a spectral representation. In this section we present the set-up and the analysis is in Section 7. Section 7 also presents conditions so that ψ n is O(M 1/2 ) close to an eigenvector of V in L 2 (dx). To replace ψ n by such an electron eigenstate is called the Born-Oppenheimer approximation, which has been studied for the time-independent [41, 7] and the time-dependent [24, 38] Schrödinger equations by different methods. To construct the solution operator it is convenient to include a non interacting particle, i.e. a particle without charge, in the system and assume that this particle moves with constant high speed dx1/dt 1 = p (or equivalently with speed one and larger mass); such a non interacting particle does clearly not effect the other particles. The additional new coordinate X1 1 is helpful in order to simply relate the time-coordinate t and X1. 1 To not change the original problem (1.1), add the corresponding kinetic energy (p 1 1) 2 /2 to E and write equation (2.8) in the fast time scale τ = M 1/2 t and change to the coordinates i d dτ ψ n = (V V n ) ψ n 1 2M G X (G 1 ψn ) (τ, X ) := (τ, X 1 2, X 1 3, X 2,..., X N ) [, ) I instead of (X 1, X 2,..., X N ) R 3N, where X = (X 1, X 2, X 3) R 3, to obtain (2.25) i ψn + 1 2(p 1 1) 2 ψn = (V V n ) ψ n 1 2M G =: Ṽ ψ n, X (G 1 ψn ) using the notation ẇ = dw/dτ in this section; note also that G is independent of X1. 1 We see that the operator V := G 1 Ṽ G = G 1 1 (V V n )G }{{} 2M X =V V n

17 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 17 is Hermitian on L 2 (R 3J+3N 1 ). The solution of the eikonal equation (2.4), by the characteristics (2.5), becomes well defined in a domain U = [, M 1/2 t] R 3N 1, in the new coordinates. Assume now the data (X, p, z ) for X R 3N 1 is (LZ) 3N 1 -periodic, then the also (X τ, p τ, z τ ) is (LZ) 3N 1 -periodic. To simplify the notation for such periodic functions, define the periodic circle T := R/(LZ). We seek a solution Φ of (1.1) which is (LZ) 3(J+N) 1 -periodic in the (x, X )-variable. The Schrödinger operator V τ has, for each τ, real eigenvalues {λ m (τ)} with a complete set of eigenvectors {p m (x, X ) τ } orthogonal in the space of x-anti-symmetric functions in L 2 (T 3J+3N 1 ), see [4]; its proof uses that the operator V τ + γi generates a compact solution operator in the Hilbert space of x-anti-symmetric functions in L 2 (T 3J+3N 1 ), for the constant γ (, ) chosen sufficiently large. The discrete spectrum and the compactness comes from Fredholm theory for compact operators and that the bilinear form v V T 3(J+N) 1 τ w + γvw dxdx is continuous and coercive on H 1 (T 3(J+N) 1 ), see [16] and Section 7.3. We see that Ṽ has the same eigenvalues {λ m (τ)} and the eigenvectors {G τ p m (τ)}, orthogonal in the weighted L 2 -scalar product v w := v w G 2 dx. T 3N 1 The construction and analysis of the solution operator continues in Section 7 with the four steps spectral decomposition, derivatives of the wave function, discrete spectrum and the Born-Oppenheimer approximation. Remark 2.1 (Boundary conditions). The eigenvalue problem (1.1) makes sense not only in the periodic setting but also with alternative boundary conditions from interaction with an external environment, as seen from the minimization formulation. The inflow, with data given from the time-independent Schrödinger problem, and the outflow of characteristics gives a different perspective on molecular dynamics simulations and the possible initial data for the time-dependent Schrödinger equation. 3. The time-dependent Schrödinger equation The corresponding time-dependent Ansatz ψ n (x, X, t)e im 1/2 (θ n(x,t)+et) in the time-dependent Schrödinger equation [39] i (3.1) M Φ = HΦ 1/2 leads analogously to the equations t θ n + Xθ n 2 = E V n, 2 t ρ + X (ρ X θ n ) = M 1/2 I(ψ n X ψ n ),

18 18 ANDERS SZEPESSY coupled to the Schrödinger equation along its characteristics X t i d M 1/2 dt ψ n (x, X t, t) = (V V n ) ψ n G X ( 2M ψ n G 1 ), (3.2) Ẍ n = X( ψ n V ψ n ) ψ n ψ, n with the same equation for (X, p, ψ, ρ, z, X p, XX p) as for the characteristics (2.4) and (2.9) in the time-independent formulation. The Ehrenfest dynamics is therefore the same when derived from the time-dependent and time-independent Schrödinger equations and the additional coordinate, introduced by a non interacting particle in the construction of the timeindependent solution in Section 7, can be interpreted as time. A difference is that the time variable is given from the time-dependent Schrödinger equation and implies classical velocity, instead of the other way around for the time-independent formulation. 4. Surface-hopping and multiple states In general the eigenvalue E is degenerate with high multiplicity related to that several combinations of kinetic nuclei energy and potential energy sum to E Φ (2M) 1 X ΦdxdX + Φ V ΦdxdX = E Φ ΦdxdX, T 3(J+N) T 3(J+N) T 3(J+N) with different excitations of kinetic nuclei energy and Born-Oppenheimer electronic eigenstates. When several such states are excited, it is useful to consider a linear combination of eigenvectors n n=1 ψ ne im 1/2 θ n =: Φ, where the individual terms solve (1.1) for the same energy E. We have (4.1) H Φ = E Φ and the normalization n n=1 ψ n 2 L 2 = 1 implies Φ L 2 = 1. Such a solution Φ can be interpreted as an exact surface-hopping model. The usual surface-hopping models make a somewhat different Ansatz with the x-dependence of ψ n prescribed from a given orthonormal basis in L(T 3J ) of wave functions of different energy and with explicit time-dependence, see [42, 41]. This section extends the Ehrenfest dynamics to multiple states and presents an example of hopping between characteristics in Ehrenfest dynamics Surface-hopping and Ehrenfest dynamics for multiple states. The characteristic (X n, p n ), the wave function ψ n, the density ρ n and the phase z n (t) = θ n (X t ) determine the time-independent wave function ψ n and the corresponding Ehrenfest approximation ˆψ n (r n ˆρ n ) 1/2 e im 1/2 ẑ n =: ˆΦ (4.2) n ˆρ n ( ˆX)d ˆX = ˆρ n ( ˆX )d ˆX dt T

19 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 19 yields an approximation to Φ; the density of state n is now a constant multiple, r n, of the one-state density ˆρ n defined in (2.23), normalized to ˆρ T 3(J+N) n dx = 1 and n n=1 r n = 1. Note that by definition the Ehrenfest states ( ˆX n, ˆψ n, ˆρ n ), n = 1,..., n, satisfying (2.19) and (2.23), are uncoupled. As an alternative to the usual surface-hopping methods, simulating hopping between characteristics based on empirical hopping-rules, we propose to solve the Ehrenfest characteristics (2.19) for multiple states n with (4.2) and then include the density weight in determining the outcome, in the usual probability sense of quantum mechanics, see Section Characteristics hopping in the Ehrenfest dynamics. Here we show that the Ehrenfest dynamics provides hopping between characteristic paths, within the same state n, due to varying velocity along the characteristic path. The density ˆρ n does not in general move along the characteristic ˆX n as seen in the following simple example in the domain R [ 1, 1] ( ˆX 1, ˆX 2 ) ˆX 1 = ˆp 1 =, ˆX 2 = ˆp 2 ( ˆX) = { (C + 2ɛ 1 ˆX2 ) 1/2, ˆX1 < (C 2ɛ 1 ˆX2 ) 1/2, ˆX1 >, with vertical characteristics. This is a solution for the potential { (4.3) V ( ˆX) ɛ 1 = ˆX2 + C/2, ˆX1 < ɛ 1 ˆX2 + C/2, ˆX1 >, with a barrier at ˆX 1 = and any constant C 2ɛ 1. For the inflow-domain the density becomes I := { ˆX := ( ˆX 1, 1) ˆX1 R} (4.4) ˆρ( ˆX) = ˆρ ( ˆX ) ˆp 2 ( ˆX) d ˆX 2 ˆp 2 ( ˆX 1, ˆX 2 ) Choose the constant C such that ˆp 2 ( ˆX 1, 1) = µ, for ˆX 1 <, then ˆp 2 ( ˆX 1, 1) = (µ 2 +4ɛ 1 ) 1/2, for ˆX 1 <. Take µ = ɛ 1/2 /2 to have (µ 2 + 4ɛ 1 ) 1/2 µ 1. Choose also ˆp 2 ( ˆX 1, 1) = (µ 2 + 4ɛ 1 ) 1/2 µ 1, for ˆX 1 >, then ˆp 2 ( ˆX 1, 1) = µ, for ˆX 1 >, so that the final density is the initial density reflected around ˆX 1 =, that is ˆρ( ˆX 1, 1) = ˆρ( ˆX 1, 1). For instance a particle almost concentrated near ˆX 1 = 1, initially at ˆX 2 = 1, becomes almost concentrated at ˆX 1 = 1, finally at ˆX 2 = 1; therefore the particle has crossed the barrier ˆX 1 = without a particle path (i.e. a characteristic) crossing the barrier. The corresponding continuity equation in the time-dependent setting, which has the same characteristics as the time-independent case, would need a boundary condition on its inflow.

20 2 ANDERS SZEPESSY domain R { 1} [, T ] R [ 1, 1] {} (instead of on I = R { 1}) and therefore the setting is different from the time-independent. 5. Computation of observables Assume the goal is to compute an observable Φ A ΦdX T 3N for a given bounded linear multiplication operator A = A(X) on L 2 (T 3N ) and a solution Φ of (4.1). We have (5.1) T 3N Φ A ΦdX = A(ψ n e im 1/2 θ n ) ψ m e im 1/2 θ m dx n,m T 3N = Ae im 1/2 (θ m θ n) (ψ n ψ m )dx. n,m T 3N We can chose the eigenfunctions {ψ n e im 1/2 θ n n = 1,..., n}, for the eigenvalue E of (1.1), to be orthogonal in L 2 (T 3(J+N) ), by the Gram-Schmidt procedure, so that when A is constant the integrals in the right hand side of (5.1) vanish for n m. The integrand is oscillatory on the local scale y R 3N, for M 1/2 X (θ n θ m ) y 1, and assume the phase gradient satisfies X (θ n θ m ) c for some positive constant c and n m. Then the integrals over T 3(J+N) of such an oscillatory function e im 1/2 (θ m θ n) ψ n ψ m with respect to a differentiable function A is small of order O(M 1/2 ): (5.2) h(x)e im 1/2 (θ m θ n) dx = M 1/2 h(x) T 3N T 3N = M 1/2 = O(M 1/2 ). Consequently we have (5.3) Φ A ΦdX = T 3N n n=1 T 3N i X (θ m θ n ) X (θ m θ n ) 2 X eim 1/2 (θ m θ n) dx ( i X h(x) X (θ m θ n ) ) e im 1/2 (θ m θ n) dx X (θ m θ n ) 2 T 3N A ψ n ψ n dx + O(M 1/2 ), in the case of multiple eigenstates, n > 1, and T 3N Φ A ΦdX = T 3N A ψ 1 ψ 1 dx

21 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 21 for a single eigenstate. We will study the approximation (5.4) A ψ n ψ n dx = A(X)ρ n (X)dX n T 3N n T 3N using the notation The factorization (2.23) g(x n, ψ n, ρ n ) := A(X n )ρ n (X). ˆρ( ˆX)d ˆX = ˆρ( ˆX )d ˆX dt dt = ˆρ( ˆX )d ˆX dt T implies that each term in the observable (5.4) can be determined by T (5.5) g( ˆX, ˆψ n, ˆρ n )d ˆX = A(X t )ˆρ n ( ˆX )d ˆX dt T 3N I T. To compute this integral requires to find an approximation ˆρ n ( ˆX ) of ρ n (X ), to replace the integrals by quadrature (with a finite number of characteristics ˆX and time steps), and to approximate the characteristics by the Ehrenfest solution ( ˆX n, ˆψ n ) T I T A( ˆX t )ˆρ n ( ˆX )d ˆX dt T t ˆX A( ˆX t )ˆρ n ( ˆX ) ˆX The quadrature approximation is straight forward in theory, although costly in practical computations. Regarding the inflow density ˆρ I n there are two situations - either the characteristics return often to the inflow domain or not. If they do not return it is reasonable to define the inflow-density ˆρ I n as an initial condition. If characteristics return, the dynamics can be used to estimate the return-density ˆρ I n as follows: let ˆX : [, T ] R 3N denote Ehrenfest nuclei positions and assume that the following limits exist (5.6) lim lim 1 M T MT MT A( ˆX t )dt = lim lim M T = lim T I M m=1 T (m+1)t mt t T. A( ˆX t ) dt T M A( ˆX t) dt T ˆρ n( ˆX ; ˆX )d ˆX, which bypasses the need to find ˆρ n I and the quadrature in the number of characteristics. A way to think about this limit is to sample the return points ˆX t I and from these samples construct an empirical return-density, converging to ˆρ n I as the number of return iterations tends to infinity. We allow the density ˆρ( ˆX ; ˆX ) to depend on the initial position ˆX ; the more restrictive property to have ˆρ( ˆX ; ˆX ) constant as a function of ˆX is called ergodicity. If (5.6) holds we say that the dynamics is weakly ergodic. For the Ehrenfest Hamiltonian

22 22 ANDERS SZEPESSY dynamics (2.22), the possible densities generated by such weak ergodicity are limit solutions of the Liouville equation t ˆρ + p X ˆρ ˆφ X V ˆφ p ˆρ M 1/2 V ˆφ i ˆφr ˆρ + M 1/2 V ˆφ r ˆφi ˆρ =, in the sense ˆρ = lim t ˆρ, which take the form ˆρ = f( p 2 /2 + ˆφ V ˆφ) for some functions f as described more in Section 6.3. The more accurate density ρ n (X) in (2.24) does not factorize ρ(x)dx ρ(x )dx dx 1 1/ p 1 1(X) and consequently ergodicity for the exact paths seems not hold in the same sense. 6. Approximation error derived from a Hamilton-Jacobi equation A numerical computation of an approximation to n ψ T 3N n Aψ n dx has the main ingredients: (1) to approximate the exact characteristics by Ehrenfest characteristics (2.19), (2) to discretize the Ehrenfest characteristics equations, and either (3) if ρ I is an inflow-density, to introduce quadrature in the number of characteristics, or (4) if ρ I is a return-density, to replace the ensemble average by a time average using the weak ergodicity (5.6). This section presents a derivation of the approximation error, avoiding the second discretization step studied for instance in [9, 1, 32] The Ehrenfest approximation error. This section shows that both alternative Ehrenfest systems, written as a Hamiltonian system (1.5) or the alternative form (2.19), approximate Schrödinger observables. We see that the approximate wave function ˆΦ, defined by (6.1) ˆΦ = ˆρ 1/2 ˆψn e im 1/2 ˆθ n, where ˆψ n = ˆφ n e im 1/2 R t ˆφ n V ˆφ n(s)ds and ( ˆX, ˆφ n ) solves the Hamiltonian system (2.21) or the system (2.2), is an approximate solution to the Schrödinger equation (1.1) (6.2) (H E)ˆΦ = 1 e im 1/2 ˆθ n X (ˆρ 1/2 ˆψn ), }{{} 2M =:γ

23 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 23 since by (2.2), (2.9), (2.17) and (2.19) (H E)ˆΦ = ( im 1/2 ˆψn (V V n ) ˆψ ) n ˆρ 1/2 e im 1/2 ˆθ n }{{} = + ( X ˆθn ˆψ n V ˆψ n E ) ˆρ 1/2 ˆψn e im 1/2 ˆθ n }{{} = eim 1/2 ˆθn 2M X (ˆρ 1/2 ˆψn ) = 1 2M eim 1/2 ˆθn X (ˆρ 1/2 ˆψn ). }{{} =:v Therefore ˆΦ approximates a non degenerate eigenstate Φ, satisfying HΦ = EΦ, or more generally the span of ˆΦ approximates the eigenspace spanned by Φ. This section presents conditions for accurate approximation error of observables g(x) ρ(x) dx g(x) ˆρ(X) dx. T }{{} 3N T }{{} 3N =Φ Φ =ˆΦ ˆΦ The expansion in the orthonormal eigenpairs {λ n, Φ n }, satisfying HΦ n = λ n Φ n, yields which establishes n ˆΦ =: n α n Φ n (λ n E)α n Φ n = 1 2M v, (λ n E)α n = 1 2M Φ n v dx. T } 3N {{} =: ˆv n Since ˆΦ exists and solves (6.2), the right hand side satisfies ˆv n =, for n such that λ n = E. Use Parsevals relation to rewrite the L 2 -norm of v We have ˆv n =, when λ n = E, and let v 2 L 2 (T 3(J+N) ) = n ˆv n(e) := lim sup δ + ˆv n 2 ˆv n (E + δ), δ

24 24 ANDERS SZEPESSY which implies Assume that (6.3) n ˆv n (λ n ) 2 λ n E 2 {n: λ n E <1} {n: λ n E <1} ˆv n(e) 2 + n ˆv n 2 = O(1), n ˆv n(e) 2 = O(1). ˆv n 2. Remark 6.2 motivates assumption (6.3), when the number of nuclei is large compared to M. Assumption (6.3) yields 4M 2 α n 2 = ˆv n 2 (6.4) (λ n n E) = O(1), 2 n : λ n E and we conclude that there exists an eigenstate Φ, satisfying H Φ = E Φ, such that Let ρ := Φ Φ and ˆρ := ˆΦ ˆΦ. We have ˆΦ Φ L 2 (T 3(J+N) ) = O(M 1 ). Theorem 6.1. Assume that the bound (6.4) holds, then Ehrenfest dynamics (6.1) approximates Schrödinger observables with the error bounded by O(M 1 ) : (6.5) g( ˆX)ˆρ( ˆX)d ˆX = g(x)ρ(x)dx + O(M 1 ). T 3N T 3N This derivation assumed that v = e im 1/2 X (ˆρ1/2 ˆψ) is bounded in L 2 (T 3(J+N) ) and used the residual, v/(2m), of the Ehrenfest solution ( ˆX, ˆp, ˆψ n, ) inserted into the time-independent Schrödinger equation (1.1) with the solution Φ; the derivation did not use the exact solution paths (X, p, ψ n ) obtained from the exact characteristics. The observable may include the time variable, through the position of a non interacting (non-charged) nucleon moving with given velocity, so that transport properties as e.g. the diffusion and viscosity of a liquid may by determined, cf. [21]. Remark 6.2. So far we did not use that the system size is large; to motivate assumption (6.3) we will use that the number of nuclei N is large compared to M, so that M/N is negligible small. We have λnˆv n = T 3N λn Φ n vdx and to estimate the derivative λn Φ n we make a perturbation δ of the total energy λ n and study its effect on the eigenfunction Φ n. When λ n = p 2 /2 + V n is perturbed by δ, we assume that this energy δ is spread approximately uniformly to all nuclei as kinetic energy, so that the momentum for each particle changes only O(δN 1 ). The assumption that the perturbation energy is spread to kinetic energy uniformly comes with a question, why uniformly and not, for instance, all energy to one particle: we know there are many eigenstates with eigenvalue λ n + δ and we pic one special that is close to Φ n by making this almost uniform distribution of kinetic energy. Then the particle paths

25 MOLECULAR DYNAMICS DERIVED FROM THE SCHRÖDINGER EQUATION 25 change negligible and the eigenvector Φ = ψe im 1/2θ has almost the same path (X, p, ψ). The phase, written as θ = 2 t λ n V n ds, can also have negligible change by considering the perturbed phase at time t, depending on the change from λ n to λ n + δ through t λ n + δ V n ds = t λ n V n ds, so that the phase θ does not change. We see that for large N the derivative λn Φ n becomes negligible small and hence λnˆv n = T 3N λn Φ n vdx is small. Section 7.2 motivates the L 2 -bound v L 2 = n ˆv n 2 = O(1), from derivatives of the electron wave function, using a spectral decomposition in L 2 (dxdx) The Born-Oppenheimer approximation. The Born-Oppenheimer approximation leads to the standard formulation of ab initio molecular dynamics, in the micro-canonical ensemble with constant number of particles, volume and energy, for the nuclei positions X, (6.6) X = X V V := ψ V ψ ψ ψ, by using that the electrons are in the eigenstate ψ with eigenvalue V to V. Oppenheimer dynamics approximates the Ehrenfest dynamics The Born- ˆX = ˆψ n X V ˆψ n ˆψ n ˆψ n. In (7.14) we show that ˆψ n = ψ + O(M 1/2 ), which yields O(M 1/2 ) difference between the two right hand sides. A more careful study of such approximations with stochastic initial data for ˆψ n ψ, depending on the temperature, is in Section 6.3. Here we apply the error analysis of the previous section to the Born-Oppenheimer approximation. The Born-Oppenheimer approximation can be analyzed similarly as the Ehrenfest dynamics by using the approximation ˆΦ = ρ 1/2 ψ e im 1/2 θ, satisfying (6.7) (H E)ˆΦ = M 1/2 i ψ ρ 1/2 1 2M eim 1/2 θ X ( ρ 1/2 ψ ), with the additional residual term including the bounded factor ψ = X ψ Ẋ the derivation in the previous section to deduce = O(1). Apply

26 26 ANDERS SZEPESSY Theorem 6.3. Assume that v = e im 1/2 θ ( i ψ ρ 1/2 (2M) 1/2 X ( ρ1/2 ψ ) ) L 2 (T 3(J+N) ) and suppose that (6.4) holds for this v, then the Born-Oppenheimer dynamics (6.6) approximates, in the weakly ergodic case (5.6), Schrödinger observables with error bounded by O(M 1/2 ) : T lim g( X t ) dt T T = g( X) ρ( X)d X T (6.8) 3N = g(x)ρ(x)dx + O(M 1/2 ). T 3N 6.3. Stochastic Langevin and Smoluchowski molecular dynamics approximation. In this section we analyze a situation when the electron wave function in the Ehrenfest dynamics is perturbed from its ground state by thermal fluctuations, which will lead to molecular dynamics in the canonical ensemble with constant number of particles, volume and temperature. To determine the stochastic data for φ requires some additional assumptions. Inspired by the study of a classical heat bath of harmonic oscillators in [44], we will sample the initial data for φ randomly from a probability density given by an equilibrium solution f (satisfying t f = ) of the Liouville equation t f + pe H E re f re H E pe f =, to the Ehrenfest dynamics (2.22). There are many such equilibrium solutions, e.g. f = h(h E ) for any differentiable function h and there may be equilibrium densities that are not functions of the Hamiltonian. As mentioned in Section 5, the set of densities ˆρ, corresponding to the different eigenstates, forms the relevant set of equilibrium solutions f. To find the equilibrium solution to sample φ from, we may first consider the marginal equilibrium distribution for the nuclei in the Ehrenfest Hamiltonian system. The equilibrium distribution of the nuclei is simpler to understand than the equilibrium of electrons: in a statistical mechanics view, the marginal probability of finding the now classical nuclei with the energy H E := 2 1 p 2 + φ V (X)φ (for given φ) is proportional to the Gibbs-Boltzmann factor exp( H E /T ), where the positive parameter T is the temperature, in units of the Boltzmann constant, cf. [17], [27]. Assuming that equilibrium density is a function of the Hamiltonian, its marginal distribution therefore satisfies h ( 2 1 p 2 + φ V (X)φ ) dφ r dφ i = e p 2 /(2T ) C(X) for some function C : R 3N R and Fourier transformation with respect to p R of this equation implies the Gibbs distribution h(h E ) = ce H E/T, for a normalizing constant c = 1/ exp( H E /T )dφ r dφ i dxdp. In this perspective the nuclei act as the heat bath for the electrons. The work [28] considers Hamiltonian systems where the equilibrium densities are assumed to be a function of the Hamiltonian and shows that the first and second law of (reversible) thermodynamics hold (for all Hamiltonians depending on a multidimensional set of parameters) if and only if the density is the Gibbs exponential exp( H E /T ) (the if part was formulated already by Gibbs [23]); in this sense, the Gibbs