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The Linearized Semiclassical Initial Value Representation (LSC-IVR) method with a Local Gaussian Approximation (LGA)
The LSC-IVR [21, 57, 22] approximation is stricly equivalent to the LPI approximation detailed before. The originality of the method develop by Liu and Miller is to combine this approximation with the Local Gaussian Approximation (LGA) [13, 20] for sampling the Wigner density. As a consequence, I will insist here on the Local Gaussian Approximation and why it is a good method to compute Wigner densities.
I remind the here the LSC-IVR approximation (or LPI) result for the standard quantum correlation function in the multidimensional case: CAB(t; ) = (2~)N Z Z dx0dp0 h e bH b A h W (x0; p0)BW(xt; pt) (2.145).
where the Wigner transform has been defined in the equation 2.126, N is the number of degrees of freedom and x0 = (x(1) 0 ; ; x(N) 0 ) (idem for p0).
Calculation of the Wigner function for operator b B in equation 2.127 is usually straightforward; in fact, b B is often a function only of coordinates or only of momenta, in which case its Wigner functions is simply the classical function itself. However, as shown in the previous section, calculating the Wigner transform of e bH b A is far from being trivial. Nonetheless, this is necessary in order to obtain the distribution of initial conditions of coordinates x0 and momenta p0 for the real time trajectories. Usually, local harmonic approximations of the potential (such as the Local Harmonic Approximation (LHA) of Shi and Geva [5], the FK-LPI of Poulsen and Rossky [1], …) are adapted and give good results to many quantum simulations. These methods, however, fail when the imaginary local frequency and temperature are such that ~j!j and so are not reliable for problems dominated by potential barriers at low temperature.
The LGA for the momentum distribution attemps to remedy this problem bymodifying the LHA of Shi and Geva [5]. First a factorization of the diagonal matrix element of the Boltzmann operator (which can be evaluated accurately by path integral techniques) is performed in the Wigner function (or density) W(x; p).
Wigner densities via the phase integration method (PIM)
In 1932, Wigner introduced his representation of quantum operators to formulate quantum statistical mechanics in a language with important analogies with the classical case. Indeed, in Wigner’s representation, the average of a quantum operator b A, defined as we have seen in Chapter 2 as Tr[b b A], where b is the density operator of the system, can be identically rewritten as an integral over momenta and coordinates variables, thus: h b Ai = Tr[b b A] = Z dqdp Aw(q; p)W(q; p).
Edgeworth expansion for the Wigner density
The manipulations presented above give, by construction, a positive definite Wigner density. While this is usually not considered a serious problem for most realistic applications (and indeed, essentially all existing approximation schemes have the same limitation), it is interesting to explore alternative representations that can reproduce also negative features of the Wigner density if they exist. To that end, let us consider again the integral over r at the left hand side of equation 3.17. It is the characteristic function of the conditional probability c(rjr). We have previously introduced a cumulant expansion of this quantity, which, for all practical purposes, will be truncated in the following at second order thus approximating this integral by a Gaussian in p, with an r dependent variance (covariance matrix in the multidimensional case). By taking this Gaussian as a reference it is then possible to trivially rewrite the integral as: Z dr e i ~ prc(rjr) = e2 2 p2 ~2 Z dr e 2 2 p2 ~2 e i ~ prc(rjr).
Structure of the PIM algorithm
Sampling P(r; p) (or P2(r; p)) is non trivial since both m(r) and eE(p;r) can only be estimated numerically. Given their expressions, we will show that however, they can be calculated as an average with an associated variance. Due to this standard Monte Carlo methods, which rely on the analytical knowledge of the density, cannot be directly applied. We are going to describe how to circumvent this problem by combining two schemes for sampling « noisy » probability densities like the one in equation 3.22. These methods are the penalty [10] and Kennedy [67] Monte Carlo algorithms. Both schemes use two main steps: (1) appropriate, unbiased, numerical estimators of the probability densities are defined to generate trial values for the random variables; (2) the trial values are accepted or rejected based on a generalized acceptance criterion which, on average, corrects for the effect of the noise. For convenience in the description of the algorithm, we introduce the following notations.
Kubo correlation functions for Infrared spectroscopy
We will derive in this section the PIM version of the Kubo correlation functions.
The first case will be for operators linear in positions (it is possible to do it for more general position operators but we will not use it). Indeed, this particular case will show us how the computational cost associated to the Kubo expression is reduced compared to the symmetrised case on the most simple test case: the position autocorrelation function of a 1D harmonic oscillator.
Linear operators in the position, as we can see on the equations 4.16, are relevant for IR spectra and are appropriate for the isolated molecules considered here. Applicationof the methods to bulk systems in periodic boundary conditions is, however, simpler starting from the dipole-derivation expression in equation 4.17. I will then discuss how we can calculate Kubo correlation functions with operators linear in momentum.
Table of contents :
1 Introduction
2 State of the art
2.1 Formalism of path integrals
2.1.1 Path integrals and the canonical density matrix
2.1.2 Polymer isomorphism and PIMD
2.1.3 Time-independent equilibrium properties in the canonical ensemble from path integral formulation
2.2 Dynamical properties from quantum correlation functions
2.3 Empirical quasi-classical methods: RPMD and CMD
2.3.1 Outline of RPMD
2.3.2 Outline of CMD
2.4 Linearized methods
2.4.1 The Linearized Path Integral (LPI) representation of quantum correlation functions
2.4.2 The FK-LPI method
2.4.3 The Linearized Semiclassical Initial Value Representation (LSC-IVR) method with a Local Gaussian Approximation (LGA)
2.5 Conclusion
3 Wigner densities with the Phase Integration Method
3.1 Wigner densities via the phase integration method (PIM)
3.1.1 The PIM expression for the thermal Wigner density
3.1.2 Edgeworth expansion for the Wigner density
3.2 Structure of the PIM algorithm
3.2.1 Structure of the algorithm
3.2.2 Definition and evaluation of the numerical estimators .
3.3 Results
3.3.1 Harmonic oscillator
3.3.2 Morse potential
3.3.3 Proton transfer model
3.4 Conclusions
4 Application of PIM to the Infrared spectroscopy
4.1 Symmetrised correlation function for Infrared spectroscopy
4.2 Kubo correlation functions for Infrared spectroscopy
4.2.1 Operators linear in positions
4.2.2 Operators linear in momentum
4.2.3 Application to the Infrared spectroscopy
4.3 Conclusions
5 Application of PIM to the calculation of rate constants
5.1 General expression for chemical rate constant
5.2 PIM Flux-side correlation function
5.3 Free energy calculations
5.4 Results
5.4.1 Free energy profiles
5.4.2 Rate constants
5.5 Conclusion
6 Conclusion
A Modification of the algorithm for the multidimensional case
B Monte Carlo sampling of the polymer chains
B.1 Staging variables
C Infrared spectroscopy
C.1 Absorption coefficient from the Fermi Golden rule
D Alternative demonstration for the Kubo momentum autocorrelation function
D.1 Notations of Kubo
D.2 Momentum autocorrelation function
D.3 Equipartition of the energy
E Eckart transformation
Bibliography
