The Feynman path integral approach
The Feynman path integral approach to quantum mechanics is different from the more common Schrödinger formulation. In particular, the quantum mechanical probability amplitude is obtained from a sum over paths rather than from a wave function. The Feynman path integral approach has an obvious connection to classical mechanics. This means that it easily lends itself to useful approximations, particularly for cases which are largely solved by classical mechanics, but where quantum effects still matter.
In principle there is an infinite number of paths to sum up in the path integral, albeit only a subset will be important. Still, an exact evaluation of the path integral, analytically or numerically, is rarely possible. A nice feature is however, that in the classical limit, which can be obtained by letting , classical mechanics is obtained. This makes the path integral approach favorable for finding approximate ways of performing quantum dynamics.
The classical limit can be seen as follows. As approaches zero, the integral in Eq. (50) becomes highly oscillatory so that all paths will cancel each other except the one where the action is at an extremum. Thus we want to find the path where a small variation of it results in no variation in the action, that is
which can be written
Using integration by parts yields
The first term on the right-hand side vanishes since at the end points. We thus have the condition
which is the classical Lagrangian equation of motion. Thus, in the classical limit, obtained by letting , only paths obeying classical mechanics survive. We have thus seen how classical mechanics is naturally embedded in the path integral approach.
In reality is small, but not zero. Thus, assuming that only the extremum path survives is an approximation. This approximation is referred to as the stationary phase approximation, since a small variation in the extremum path gives no change in the associated phase.
The Feynman path integral is either solved in real time or in imaginary time. Ordinary physical processes occur in real time, which we shall look at first. In terms of the time evolution operator
the propagator can be written
in the position eigenstate basis. We now split into N equal time slices and write
By inserting the resolution of identity
N + 1 times in the time evolution operator we get
Collecting all time slices, we get
The exponential is purely imaginary and the resulting phase oscillations make the evaluation of the integrals in Eq. (66) extremely difficult, which is referred to as “the sign problem.”
Equilibrium structures and partition functions can be obtained by considering the path integral in imaginary time. Such path integrals have the huge advantage of not suffering from the sign problem. Therefore, problems involving hundreds of degrees of freedom can be solved when working in imaginary time.
Integrating out the momentum variables in Eq. (69) yields
As the coordinates are cyclic, the path that they represent is referred to as a ring polymer, or a necklace, and each coordinate position on it is referred to as a bead. The beads are connected to their nearest neighbors by harmonic springs and Veff can be seen as an effective potential. The original quantum problem has been mapped onto a closed chain polymer problem.
The average value of a position dependent operator can be obtained as
The multidimensional integrals in Eq. (72) can be evaluated by Monte Carlo sampling the positions, resulting in the path integral Monte Carlo (PIMC) method.[117, 118] For large enough values of N and sufficient number of Monte Carlo samples, converges to the exact value.
An alternative to evaluating by PIMC is to use the path integral molecular dynamics (PIMD) method. PIMD was introduced by Parrinello and Rahman in a numerical investigation of the properties of an electron solvated in molten KCl. In PIMD, a classical Hamiltonian is obtained by adding momenta to the effective potential and then performing molecular dynamics. This may for N beads, still in one dimension, be expressed as
The masses used in the original PIMD calculations could be arbitrarily set as the momentum terms were added just to explore the potential energy surface and the investigated static properties were independent of the masses. A thermostat may be used to control the temperature. The thermostat acts by scaling the velocities so that the temperature as calculated from the kinetic energy corresponds to the desired temperature. Popular thermostats are the Berendsen et al. and Nosé–Hoover thermostats.[122, 123]
Ring polymer molecular dynamics
Many dynamical quantities can be calculated from real-time correlation functions. The standard form for a real-time correlation function is
and is the ordinary Hamiltonian operator. An alternative form is the Kubo-transformed correlation function
which has more classical-like properties than the standard form in Eq. (74). The Fourier transform of the Kubo-transformed correlation function,
is related to the Fourier transform of the standard correlation function,
so that either one can be obtained form the other and they are identical for . Diffusion coefficients for example can be calculated as
where is the velocity–velocity autocorrelation function and is the Kubo-transformed one.
RPMD[24, 124] uses the same Hamiltonian form as PIMD, given in Eq. (73). This Hamiltonian differs from the ordinary Hamiltonian in the addition of the term which is a harmonic potential with temperature (β) dependent spring constants. The difference between PIMD and RPMD is that in RPMD the masses used are the physical masses and rather than using the molecular dynamics as a sampling tool, the dynamics is interpreted literally and dynamical quantitates are evaluated from the Kubo-transformed correlation functions. Approximately evaluating the relevant Kubo-transformed correlation function is the basic task in RPMD. Craig and Manolopoulos have shown that when applying RPMD, it is more appropriate to work with Kubo-transformed correlation functions than the standard ones. This relates to the more classical like properties of the Kubo-transformed correlation functions and that RPMD is exact in the classical limit.
RPMD has turned out to be very useful for calculating many properties, including thermal rate constants. The experience is that RPMD gives the accurate thermal rate constant (for a given potential energy surface) to within a factor of two or three.[125, 126] Richardson and Althorpe were able to explain how RPMD relates to semiclassical instanton theory and also show that in the deep tunneling regime the RPMD rate constant will typically underestimate the true rate constant for symmetric barriers and overestimate it for asymmetric barriers. A version of so-called centroid molecular dynamics (CMD) can be implemented in a similar way to RPMD (partially adiabatic CMD). The main difference to RPMD lies in the choices for the masses. CMD uses much smaller bead masses than RPMD (except for the centroid). Related to this, CMD is preferred over RPMD for calculating spectra.[128, 129] This is due to the extra frequencies resulting from the harmonic springs between the beads which affect the RPMD calculated spectra, whereas in CMD this can be avoided by the choice of masses.
We shall not do it here, but it can be shown that RPMD is exact
- in the high temperature limit
- in the short time limit
- for the harmonic case, provided that or is linear in x.
It commonly occurs in chemical reaction dynamics that quantum effects are present and should be treated, but still classical mechanics would be a good starting point. As RMPD is exact in the high temperature limit, that is in the classical regime, it can be expected to work well also close to this regime. In cases where the correlation function decays quickly, for instance in condensed phase, RPMD is also expected to perform well since it is exact in the short time limit. Finally, there are many situations when anharmonic effects should be treated, but the harmonic case would still be a reasonable approximation. Thus, from the third itemized statement earlier we would expect RPMD to perform well also in such situations.
In RPMD, Eq. (76) is approximately evaluated as
and subscript “0” indicates time t = 0.
In the calculation of thermal rate constants, the Kubo-transformed flux-side correlation function
can be used. It has the same long-time limit as in Eq. (11), whereby the thermal rate constant is obtained as
We shall not dwell further on the details of RPMD implementations but to mention the program “RPMDrate” for calculating gas phase rate constants of bimolecular reactions. This code uses the Bennett–Chandler procedure[131, 132] to obtain the rate constant as the product of a transmission coefficient and a centroid density quantum transition state theory rate constant. In this way, the absolute values of the partition functions are not needed, which is an important advantage.
In Table 4, some calculated thermal rate constants for the reaction are shown. Results from MCTDH and RPMD calculations are shown as well as results from canonical variational transition state theory with microcanonically optimized multidimensional tunneling[8, 7] and classical transition state theory (TST). The potential energy surface that has been used to calculate all rate constants in the table is due to Espinosa-Garcia. The rate constants from the MCTDH calculations are considered to be the most accurate ones, against which the others can be compared. It is estimated that the MCTDH calculations are converged to within about 10%, meaning that the MCTDH calculated rate constants should be within 10% of the exact values for the given potential energy surface.
The MCTDH rate constants are by far the most time consuming to obtain in Table 4 and the calculations could only be numerically converged over the temperature range from 225 to 400 K. The calculations are, however, very useful as benchmarks against which other results can be compared. Note that such comparisons should always be done on the same potential energy surface, which also explains why it is preferable to validate an approximate theory against an accurate theory rather than against experiment.
From Table 4 it can be seen that in the compared temperature range, the RPMD rate constants are within a factor of two of the MCTDH ones. The rate constants obtained from TST are much too small at low temperature since tunneling is not accounted for. The TST results are, however, much more accurate at higher temperature where tunneling plays a much smaller role. It should also be remembered that recrossing is left out in TST, which generally has the effect of making the TST rate constants somewhat too large.
It is noteworthy that the canonical variational transition state theory calculations with microcanonically optimized multidimensional tunneling in Table 4 are in good agreement with the MCTDH and RPMD results. It is suggested that in cases where only thermal rate constants are desired, transition state theory, with a tunneling correction when relevant, should always be considered before indulging in more demanding calculations.