Real time quantum dynamics simulation of large molcular systems present a challenge to theoretical physics and chemistry. Indeed, computer simulation of time-dependent quantities for quantum systems is hindered by the exponential scaling of exact methods with the number of degrees of freedom. However, access to these quantities is crucial in many interesting problems and in particular to compute, via linear response theory and time correlation functions, the response of the system to external perturbations. The development of approximate quantum dynamical schemes is therefore an active area of research. One of the main direc-tions currently pursued is to adapt efficient techniques for classical simuladirec-tions, based on molecular dynamics (MD) and/or Monte Carlo, to the quantum case. A reasonable compromise among accuracy and computational cost is realised by the so-called quasi-classical methods for computing time correlation functions. In these methods, sampling of the thermal equilibrium density for the system is combined with generalised trajectories that provide an approximation for quantum dynamics usually valid for short times.
In this section, we will present different quasi-classical methods, containing different approximations. First, we will introduce the different forms of quantum correlation functions which is possible to compute. These expressions are equivalent in the sense that they can be related to one another analytically and therefore have the same physical content. It is however useful to discuss them all since, depending of the adopted approximation, some may be more advantageous from a computational point of view. Then, we will introduce two of the most popular quasi-classical methods, which, although not formally justified, have the advantage of converging with a small number of trajectories: centroid and ring polymer molecular dynamics (CMD and RPMD, respectively) [2, 29, 30, 31, 32, 33]. Finally, we will describe three methods, which are part of the category of the linearized quasi-classical meth-ods. The Feynman-Kleinert linearized path integral (FK-LPI) method introduced by Poulsen and Rossky [1, 34, 35, 36, 37, 38], the linearized semi-classical initial value representation (LSC-IVR) with the local Gaussian approximation (LGA) describe by Liu and Miller [4,13,19,39,40,41] and finally our method, the phase integration method (PIM) introduced by Bonella and Ciccotti [7,8,9,12,17,18].
Time correlation functions play a crucial role in relating macroscopic obser-vations accessible in experiments to the microscopic dynamics of physical systems.
Although the statstical mechanics required to establish this link, in particular linear response theory, is valid for classical and quantum systems, our ability to perform calculations differs dramatically in the two situations as explained before.
Furthermore, for quantum systems, different definitions of correlation functions are possible due to the fact in quantum statistical we deal with operators which do not commute.
The first and most common possibility is to evaluate the standard quantum expression of the time correlation function:
CAB(t, β) = 1 ZT rh
e−βHbA(0)b B(t)b i
(2.64) where β = k1
BT, Z is the canonical partition function (defined in the previous section) and A(0)b and B(t)b are Heisenberg-evolved quantum operators at time 0 and time t:
B(t) =b eitH/~b Beb −itH/~b (2.65)
In the next sections, during the description of the different methods, we will explain and show how the dynamical part is approximated.
In the equation 2.64, the correlation function is written in the standard way as the thermal average of the operator product A(0)b B(t). However, this is not the onlyb possibility. A more symmetric alternative introduced by Kubo [42, 43, 44] is the Kubo-transformed correlation function:
KAB(t, β) = 1 βZ
Z β 0
dλ T rh
e−(β−λ)HbA(0)eb −λHbB(t)b i
(2.66)
In the equation 2.66, the Boltzmann operator is averaged between A(0)b and B(t).b This differs from CAB because the operators Aband Bb do not necessarly commute withH. This form of the quantum correlation is the closest to the classical one andb as a consequence is well suited to the semi-classical approximations.
A last alternative form of the quantum time correlation function has been proposed by Schofield in 1960 [45] originally for the neutron scattering. Schofield0s function, also known as the symmetrized correlation function, is defined as:
GAB(t, β) = 1 ZT rh
Aeb it∗cH/b ~Beb −itcH/b ~i
(2.67)
wheretc=t−i~β/2. In the equation 2.67, the Boltzmann and time operators are equally distributed between the operatorsAbandBb.
Schofield0s form of the correlation function shares some formal properties with clas-sical correlation functions, it is for example a real function by construction. Thus suggests that it could be a good starting point for describing semiclassical sys-tems [21,46,47,3,48,36].
These three different definitions of quantum correlation function are linked together
via their Fourier transform. Indeed, we have: where the time Fourier transform is:
F˜(ω) = Z +∞
−∞
dt e−iωtF(t) (2.69)
To obtain the relations in equation 2.68, we have to evaluate the traces in the different definitions in the basis of energy eigenstates. As an example, we carry out the calculation for the standard and the symmetrized correlation function. We have:
CAB(t, β) =1
where |ni is an eigenvector in the energy basis andEn an eigenvalue in this basis, soH|nib =En|ni. Then it follows after inserting the identity operator in the energy basis between the operators Aband Bb (these operators are usually not diagonal in this basis set):
If we proceed in the exact same way for the symmetrized correlation function we obtain: Then we take the Fourier transform of the equation 2.70which leads us to:
C˜AB(ω) = 1
where we used:
and for the Schofield0s form of the correlation function we have:
G˜AB(ω) =1
It is now straightforward to see that the relation in the equation2.68is verified:
C˜AB(ω) =eβ~ω/2G˜AB(ω) (2.76)