This article is focussed on describing some quantum dynamical approaches that are suitable for studying bimolecular chemical reactions with a focus on obtaining the thermal rate constant. The quantum dynamical time-dependent wave packet (TDWP) approach is described in some detail including a brief discussion of the multiconfigurational time-dependent Hartree (MCTDH) approach. Time-independent coupled-channel (CC) calculations are briefly described as well as the path integral based ring polymer molecular dynamics (RPMD) method. Classical and mixed quantum-classical dynamics approaches are left out.
Thermal rate constants are central to chemistry so it is not surprising that several ways to obtain them exist. Most chemical reactions occur with an activation energy, the magnitude of which is most important for the thermal rate constant. The importance of the barrier height also implies that if its value is inaccurate, a large error in the calculated rate constant is to be expected. If we for simplicity assume an Arrhenius behavior of the rate constant, it is simple to calculate how an error in the barrier height affects the rate constant. This is illustrated in Table 1, from which it is clear that having an accurate barrier height becomes exceedingly important at low temperature. It is also clear that if the error in the barrier height can be reduced by for instance very accurate electronic structure calculations, this is of great help in improving the calculated rate constant.
Table 1. Errors in the thermal rate constant for various errors in the barrier height of the potential energy surface.
T(K)
1 kcal/mol error
0.25 kcal/mol error
0.1 kcal/mol error
The results show by which factor the thermal rate constant is in error and have been obtained by assuming an Arrhenius behavior.
300
5.3
1.5
1.2
200
12
1.9
1.3
100
150
3.5
1.6
50
22,000
12
2.7
20
72,000,000
520
12
Chemical reaction dynamics is usually performed on a given potential energy surface. The Born–Oppenheimer approximation is thus normally invoked. This allows potential energy data to be obtained with an electronic structure program for fixed nuclear geometries. Analytic functions are usually fitted to the data so as to obtain a potential energy surface. The dynamics can then be performed using classical equations of motion or quantum mechanically based equations. Much work is also done for reactions which involve more than one potential energy surface, but we shall not be concerned with that here.
An alternative to perform the dynamics on an analytic potential energy surface is to use direct dynamics. In direct dynamics, the potential energy is obtained during the simulation (on the fly) rather than from a predefined analytic potential energy surface. Direct dynamics comes in different forms,[1-4] Born–Oppenheimer direct dynamics and extended Lagrangian methods including Car–Parrinello molecular dynamics[5] and its extensions (e.g., the atom-centered basis functions and density matrix propagation approaches), and nonadiabatic methods based on multiple potential energy surfaces. In these methods, the atoms move according to classical equations of motion and I will therefore not discuss them further. There has been (at least) one successful attempt[6] to perform what can be called direct quantum dynamics in the sense that an analytic potential energy surface is not created. This, however, is extremely CPU intensive and not likely to be commonly done.
Classical dynamics calculations have a number of advantages over quantum mechanical methods. They are generally
easier to implement,
much faster to perform, and
can handle substantially larger systems.
Classical methods often give surprisingly good results, which is particularly true when the quantities of interest are averages over more detailed quantities. Thus, when applicable, typically when many quantum states are populated, classical dynamics is the first choice rather than quantum mechanical methods. Conversely, quantum mechanical methods have their definite advantages.
Accurate quantum methods can be used as benchmarks against which approximate methods can be compared and validated or rejected. Also, there are many cases where tunneling, resonances, and interference effects are important. These effects are not present in classical dynamics, thus, in such cases quantum mechanical approaches are required. Further, when state-to-state calculations are performed quantum mechanical methods may be necessary. In brief, quantum mechanical methods should be used when they are necessary for solving the problem at hand.
To illustrate how important tunneling can be, Table 2 shows the thermal rate constant for the reaction
H+CH4→H2+CH3(R1)
for a set of temperatures. The calculations were performed using harmonic quantum transition state theory (HQTST),[7] which accounts for tunneling approximately but yet fairly accurately. The HQTST rate constants are compared against those from transition state theory without tunneling treatment. From Table 2, it can be seen that for reaction R1 it is vital to treat tunneling if an accurate rate constant is to be obtained, particularly at low temperature. This is primarily due to the facts that it is a light atom (hydrogen) that is involved in the bond breaking and bond forming and that there is a substantial barrier to reaction, the height of which is 12.9 kcal/mol on the potential energy surface that was used.[8] Note also that the barrier height is not the sole factor determining the rate constant, also the shape of the barrier makes a difference as soon as tunneling matters.
Table 2. Approximate tunneling contribution to the thermal rate constant for the H+CH4→H2+CH3 reaction for various temperatures.
T(K)
Tunneling factor
The tunneling factor shows how much too small the rate constant would be if tunneling was neglected. The results have been obtained using transition state theory with and without tunneling.[7]
300
3
200
40
100
10^{8}
50
10^{32}
20
10^{80}
A second example of how important quantum effects can be in chemistry is provided by Figure 1. Here cross-sections for the formation of a CN molecule from the collision between C and N atoms are shown as a function of collision energy.[9] For this process to be possible energy must be removed from the system so that bound states of the CN entity can be accessed. This happens by emission of a photon and the process is referred to as radiative association. Forming a diatom through radiative association is a highly unlikely process, which explains why the cross-sections in Figure 1 are so small.
There are numerous spikes in the cross-sections in Figure 1, so-called resonances, which in this particular case are shape resonances. Shape resonances result from quasibound states and are due to the shape of the potential. The wave function for such a state has a large amplitude inside a potential barrier, a barrier in the real potential or due to centrifugal forces, and smaller but nonvanishing amplitude outside. These states thus remind of bound states in the sense that they have a large probability density in a limited region of space. They differ from bound states in that they have nonvanishing probability out to infinity and an energy that allows dissociation. Resonances do not show up in classical calculations as seen by the baseline in Figure 1, which comes from a calculation based on classical mechanics.[9]
Although there are numerous occasions when quantum effects are essential to the behavior of a molecular system, I will let the two examples just given suffice as motivation for the need for quantum dynamics. For the rest of this article I restrict myself to quantum mechanical approaches to reaction dynamics.
Overview of Quantum Reaction Dynamics
To briefly mention some of the historical highlights of quantum reaction dynamics I begin with the seminal article by George Schatz and Aron Kuppermann on the
H+H2→H2+H(R2)
reaction, for which differential cross-sections as well as thermal rate constants were presented in 1976.[10] This was the first time where three atoms in a chemical reaction were treated in a full-dimensional quantum dynamical study, that is, where all three vibrational degrees of freedom were treated by quantum dynamics. There had been much previous work, including several reduced dimensionality calculations, that is, where not all degrees of freedom had been explicitly treated by quantum dynamics, leading up to the full-dimensional quantum dynamical calculation of the thermal rate constant, see Ref. [10] and references therein.
There are several aspects that make R2 the computationally easiest chemical reaction there is to study by quantum dynamics. It only involves hydrogen atoms, which are light so that only few rovibrational quantum states are populated during the reaction. Due to the symmetry, reaction R2 is thermoneutral and it has just as many reactant as product states that need to be included in the calculations. As the calculations generally become more demanding, the more states that have to be treated, this is an important advantage. In an exothermic reaction, the product side is likely to have many states accessible at a given energy even though perhaps only few are accessible on the reactant side, and vice versa for an endothermic reaction. The symmetry in itself further facilitates the calculations and together this explains why it took about a decade before even an isotopic substitution in reaction R2 could be successfully treated in a full-dimensional quantum dynamical study, see for example, Refs. [11, 12].
Moving from three to four atom reactions represents a huge increase in complexity. In an exact calculation involving four atoms, six, instead of three, vibrational degrees of freedom have to be treated. Imagine expanding the wave function in a basis set for each vibrational degree of freedom and then forming the complete wave function as a sum of direct products of the basis functions for each degree of freedom. If we were to use 10–100 basis functions for each vibrational degree of freedom this would give 103−106 basis functions for a three atom reaction and 106−1012 for a four atom reaction. There are ways to improve on this, but it illustrates the basic problem that occurs when increasing the number of atoms to be treated.
The reaction
H2+OH→H+H2O(R3)
became the prototype for theoretical treatments of four atom reactions. The first full-dimensional quantum dynamical calculation for this reaction was reported in 1993 by Manthe et al.[13] A most important aspect for how this could be accomplished was that the asymptotic regions were not included in the calculations. Only the interaction region of the reaction was treated, which required a much smaller grid than would otherwise have been the case. This also meant that state-to-state reaction probabilities could not be obtained but only cumulative reaction probabilities, that is reaction probabilities summed over all product and reactant states, see more in Section “Thermal Rate Constant.” Further, these calculations were only performed for a total angular momentum of zero. The goal of the article, to obtain an accurate thermal rate constant for the given potential energy surface, could still be achieved using the so-called J-shift approximation, which is explained in Section “Thermal Rate Constant.”
The next larger chemical reaction to be studied quantum dynamically in full dimensionality was reaction R1, which was achieved in the year 2000.[14] Again cumulative reaction probabilities were obtained and converted to thermal rate constants using the J-shift approximation. In this case, there are 12 vibrational degrees of freedom to treat and the calculations were possible due to the relative simplicity of this particular six atom reaction and the use of the MCTDH method, which is numerically efficient. I will return to it in Section “The MCTDH method.” As of today, reaction R1 has been treated in full-dimensionality obtaining initial state-resolved reaction probabilities.[15, 16]
Let us now move on to have a first look at how quantum dynamics calculations are carried out. Quantum reaction dynamics is usually performed by solving either the time-independent Schrödinger equation (TISE)
ĤΨ=EΨ(1)
or the time-dependent version (TDSE)
−ℏi∂Ψ∂t=ĤΨ.(2)
If the Hamiltonian is time-independent, it is a matter of numerical convenience which of the two equations is solved in an exact calculation as they give the same result, provided that the calculations are properly performed. It is however, also the case that depending on the problem studied, one of the two approaches may be numerically more suitable and easier lends itself to useful approximations.
Early quantum reaction dynamics calculations were often performed by solving the TISE. With growing complexity of the reactions studied, solving the TDSE using wave packets became more popular. This is due to the better scaling of CPU time with the size of the problem for the wave packet approach. Typical implementations for solving the TISE would scale as O(N3), whereas the wave packet approach would scale roughly as NlnN, where N is the number of basis functions used. The importance of the scaling is illustrated in Table 3. In Section “The MCTDH method,” it will be discussed how the scaling of the time-independent approach can be improved.
Table 3. The scaling problem.
N
100
1000
10,000
100,000
1,000,000
It is assumed that it takes 1 s CPU time to solve a problem with 100 basis functions (N). The other values in the table are obtained by assuming NlnN, N^{2} or N^{3} scaling.
NlnN
1 s
15 s
3 m
40 m
9 h
N^{2}
1 s
100 s
3 h
2 w
4 y
N^{3}
1 s
17 m
2 w
40 y
40 000 y
Due to the unfavorable scaling of CPU time with increasing complexity of the reaction, numerically exact full-dimensional calculations cannot be expected to be able to handle large reactions in the near future. Reduced dimensionality calculations are one way to proceed to treat more complicated reactions. There have been many reduced dimensionality studies of reactions R1, R2, R3, and others, see for instance Refs. [17, 18] and references therein. It can be concluded that for direct reactions with barriers, the most important modes to treat explicitly by quantum dynamics are first of all the breaking and forming bonds, but for reaction R1, for example, also the umbrella motion is important. To get the thermal rate constant correct it is also important to account for the zero point energy of the “spectator modes”, that is, of the modes not explicitly treated (by quantum dynamics). In this way the correct vibrationally adiabatic ground state barrier height can be obtained, that is, the reaction barrier with the zero point energy accounted for.
There are many chemical reactions that occur in a regime where quantum effects are important but classical dynamics still is a good approximation. A way forward in such cases may be to use methods based on classical mechanics but incorporating quantum effects approximately. The Feynman path integrals[19] are then a natural starting point. Two popular approaches that are based on Feynman path integrals are the Classical Wigner[20-23] and RPMD[24, 25] methods. In the Classical Wigner method classical trajectories are run, but the initial conditions are obtained from a quantum mechanical distribution of momenta and coordinates in phase space. Also in RPMD classical trajectories are run, but using an extended phase space so that each atom is represented by a necklace of beads. The method uses the imaginary-time path integral formalism, which draws on the exact equilibrium mapping between a quantum mechanical particle and a classical ring polymer.[26] RPMD has been developed by Manolopoulos and coworkers[25] and has turned out to be very useful for calculating thermal rate constants. I will therefore return it to in Section “Ring polymer molecular dynamics.”
As already mentioned, I will focus on the thermal rate constant and give less attention to the more detailed quantities, like differential and integral cross-sections or state resolved quantities. In the next section, it is shown how thermal rate constants can be obtained, for instance using time correlation functions. Using these expressions opens avenues for efficient calculation of the thermal rate constant. Using time correlation functions makes it natural to use the time dependent versions of quantum mechanics. I will therefore be rather brief on the time-independent approach but more detailed on time-dependent calculations.
Thermal Rate Constant
The thermal rate constant of a bimolecular reaction is central to chemistry and we shall devote some extra attention to it. A route to calculate the thermal rate constant is to first obtain all state-to-state reaction probabilities. The probability, Pi→f for a transition from an initial state i to a final state f can be obtained from the so-called scattering matrix, the S-matrix, according to
Pi→fJ(E)=|SifJ(E)|2,(3)
where SifJ is an element of the S-matrix, which depends on the energy E. The S-matrix is obtained from quantum dynamical calculations, see Section “The CC method.” The superscript J indicates that the calculations are performed for a certain value of the total angular momentum.
State-to-state reaction cross-sections, σi→f(Et), for a translational energy E_{t}, are obtained by summing over the reaction probabilities for all relevant angular momenta. This means that the calculations are performed for all values of the total angular momentum that yield nonzero reaction probabilities at that energy. Thus
σi→f(Et)=πki2∑J(2J+1)Pi→fJ(E),(4)
where ki is the wave vector for the initial state and ki2=2μEt/ℏ2 with µ being the reduced mass between the reactants. Et=E−Ei, where E_{i} is the energy of the initial state. State-to-state rate constants can be obtained by Boltzmann averaging the state-to-state reaction cross-sections over energy as
where T is the temperature and k_{B} is the Boltzmann constant. Thermal rate constants can then be calculated as
k(T)=∑f∑iki→f(T)e−Ei/kBT∑ie−Ei/kBT.(6)
It should also be mentioned that from the S-matrix (obtained for all relevant J-values), all observables can be calculated.
A major drawback of the above procedure for obtaining the thermal rate constant is that the state-to-state reaction probabilities are required and to obtain these, the calculations need to cover the asymptotic regions. Thus, large grids are required, which in high dimensionality easily becomes prohibitive. If however, only the thermal rate constant is of interest, it is not necessary to calculate the state-to-state reaction probabilities. It is only necessary to know whether a collision leads to products or not. This is determined in the interaction region and it is therefore not necessary to include the asymptotic regions in the calculation.
The individual reaction probabilities can be summed over all open states, that is, the energetically accessible states at the given energy E, to give
NJ(E)=∑i,fPi→fJ(E).(7)
By summing NJ(E) over the total angular momentum, the cumulative reaction probability N(E) is obtained as
N(E)=∑J(2J+1)NJ(E).(8)
N(E) is conveniently related to the thermal rate constant[27, 28] by the expression
k(T)=[2πℏQr(T)]−1∫−∞+∞N(E)e−E/kBTdE,(9)
where Qr(T) is the reactant partition function per unit volume. Qr(T) is often evaluated in the harmonic oscillator—rigid rotor approximation. This is not always accurate enough. A useful improvement can then be to calculate the zero point energy accurately but retain the harmonic approximation for the remaining vibrational energy levels.[29]
To obtain the cumulative reaction probability, or the reaction cross-section, it may be necessary to perform scattering calculations for hundreds of J-values before the angular momentum results in a centrifugal barrier that is large enough to prevent reaction. The calculations become more demanding as the value of J increases. For reactions with substantial barriers, the J-shift approximation[30, 31] is however, often quite accurate and can be used to avoid the J>0 calculations. This was done by Manthe et al.[13] in the calculations mentioned in Section “Overview of Quantum Reaction Dynamics” for reaction R3 and similarly for reaction R1.[14]
The J-shift approximation is usually implemented so that scattering calculations are only performed for J = 0 and the reaction probabilities for all other J-values are obtained from the J = 0 probabilities by shifting them in energy. The J = 0 probabilities are shifted by an amount that corresponds to the overall rotational energy of the complex at the transition state geometry, or equivalently, the shift in effective barrier height caused by the overall rotation. The physical assumption behind the J-shift approximation is that the translational energy at the transition state determines the reaction probability. In the J-shift approximation the rate constant may be written
k(T)=Qrot‡(T)2πℏQr(T)∫−∞+∞NJ=0(E)e−E/kBTdE,(10)
where Qrot‡(T) is the rotational partition function at the transition state.
Miller et al.[32] derived expressions for the thermal rate constant in terms of time correlation functions. These expressions are computationally favorable for obtaining the thermal rate constant, just like the cumulative reaction probability, to which they can be related, see Section “The MCTDH method.” The following three correlation functions
cfs(t), css(t), and cff(t) are referred to as the flux-side, side-side, and flux-flux correlation functions, respectively. Equation (14) shows how these correlation functions are related to each other. The flux operator, F̂, appearing above is defined as
F̂=iℏ[Ĥ,ĥs].(15)
The Heaviside step function is zero on the reactant side of a dividing surface and unity on the product side. These correlation functions form the basis of the more recently developed methods that are used for calculating thermal rate constants that go beyond the zero time limit. I shall below return to how the correlation functions are utilized in the MCTDH method and in RPMD.
Time-Independent Quantum Scattering
Solving the TISE results in a boundary value problem, which is usually handled by expanding the wave function in basis functions for the degrees of freedom (coordinates) that are bounded (have a finite extent). Varying the coefficients in the expansion so as to minimize the energy leads to a (large) matrix that has to be diagonalized. For the remaining coordinate(s), the wave function is obtained by a propagation in configuration space. This is the main idea of the CC, or close-coupling, approach,[17, 33-37] which is the most popular time-independent approach to quantum dynamics. The CC equations are derived below and then some numerical aspects are given on the calculations including an application to the reaction Cl+CHD3→HCl+CD3. The Hamiltonian is assumed to be time-independent. For more details see for instance the reviews [17, 18].
The CC method
The basic idea in the CC method is to select a scattering coordinate, ρ, corresponding to a radial-like motion, and writing the Hamiltonian as
Ĥ(ρ,r)=−ℏ22μd2dρ2+ĤS(r;ρ),(16)
where r denotes all coordinates except the scattering coordinate and µ is a reduced mass for the colliding system. ĤS(r;ρ) is referred to as the surface Hamiltonian as it can be viewed as representing a (multidimensional) surface at various values of ρ.
The scattering wavefunction Ψ(ρ,r) may be expanded in a complete set of (orthonormal) basis functions Φj(r), called the channel functions. We can write
Ψ(ρ,r)=∑jψj(ρ)Φj(r),(17)
where ψj(ρ) is the population amplitude for Φj(r) and j contains all relevant labels.
Equation (17) can be substituted into the TISE, multiplied by Φi*(r) and integrated over the variables r. This results in a set of coupled second-order linear differential equations,
d2ψi(ρ)dρ2=∑j[Zij(ρ)−k2δij]ψj(ρ)(18)
where k is the wave number, that is the magnitude of the wave vector, and δij is the Kronecker delta. Z(ρ) is called the coupling matrix and its elements are given by
Zij(ρ)=2μℏ2〈Φi(r)|ĤS(r;ρ)|Φj(r)〉.(19)
Solving Eq. (18) subject to the relevant boundary conditions gives the radial functions ψj(ρ).
The CC equations are solved by numerical propagation in the scattering coordinate. Popular propagators are R matrix methods[38-40] and log-derivative propagators.[41-44] In these methods, the R matrix,
R(ρ)=ψ(ρ)ψ′(ρ),(20)
or its inverse Y(ρ)=R−1(ρ), is propagated in small steps, called sectors, from a small value of ρ out to a large value of ρ, where the interaction potential vanishes. If the sectors are small enough that the potential hardly changes within a sector, the propagation becomes analytic within each sector.
The S matrix is obtained by matching the wave function and its derivative with the analytically known asymptotic solutions. The details of this depend on the chosen coordinate system.[45-47] By calculating the S matrix for a sufficient number of values of the total angular momentum and total energies, all observable properties can be obtained,[17] such as reaction probabilities [Eq. (3)], differential cross-sections, and thermal rate constants [Eq. (6)]. There are program packages available for scattering calculations based on the mentioned propagators, like MOLSCAT,[48] HIBRIDON,[49] and ABC.[50]
The CC approach has some attractive features:
the whole S-matrix and thus all state-to-state reaction probabilities can be obtained in a single calculation.
Highly accurate results can be computed, especially at low energies. The CC approach has therefore played an important role in ultracold collision studies[51-55] and in astrophysics.[56]
It is efficient for studying complex-forming reactions, which last much longer than direct reactions.
The calculations can be performed such that using saved information, it is cheap to repeat the calculations at another energy and/or total angular momentum. As a result, detailed information, such as differential cross-sections, can be computed efficiently.
There are also computational drawbacks with CC calculations. The speed of such calculations is limited in particular by the need to obtain the functions Φ(r) in Eq. (17), which involves a matrix diagonalization.[18] The functions Φ(r) are obtained in each sector, which typically means that this is done for a few hundred values of the scattering coordinate. Another time-consuming task is to obtain the R- or Y-matrix, which requires a matrix inversion. This matrix inversion has to be performed a few hundred times for each energy and total angular momentum of interest.
Numerical aspects and application
The numerical bottle neck in CC calculations is the repeated diagonalization of a large matrix, which has to be done for each sector used in the propagation. Matrix diagonalization generally scales roughly as O(N3), where N is the size of the matrix. Here N represents the number of basis functions used to expand the wavefunction in the bounded degrees of freedom.
To alleviate the diagonalization issue, we can try to make the matrix as small as possible by finding a compact basis. This can be achieved by sequential diagonalization and truncation.[57-60] In this approach, the final matrix is built from basis functions that have already been optimized. This optimization can for instance be achieved using a large basis in one-dimension, diagonalizing the corresponding matrix, and keeping only the lower eigenstates, as these should be more relevant to use in the final diagonalization of the full problem. Additional dimensions may be added, one after the other. The direct diagonalization approach is very efficient for dense matrices with a rank of up to about 8000. Also, there are direct diagonalization algorithms for symmetric matrices,[61-63] which can achieve a CPU scaling of about O(N2.4).
In quantum reaction dynamics, only a small fraction of the eigenstates of the matrix is needed. In such cases, the unfavorable scaling of direct diagonalization algorithms can be reduced using iterative diagonalization methods[58, 64-66] that are efficient for solving the eigenvalue problem of large sparse matrices. Several such algorithms are available, including classical orthogonal polynomial expansions,[64] Davidson iteration[67, 68] and Krylov subspace iterations like the well-known Lanczos method.[69] Iterative diagonalization methods converge the eigenvalues one after the other by repeatedly acting with the Hamiltonian operator on a vector representing the wave function. These methods are therefore most effective if only a fraction of all eigenvalues are required.
The Lanzcos method[69] is known to converge interior eigenvalues in dense regions of the spectrum slowly. By a spectral transform, the eigenvalue spectrum can be transformed so that the eigenvalues of interest become less dense and are converged faster. The spectral transform is achieved by acting with f(Ĥ) rather than Ĥ on the wave function, where f is an analytic function that is expanded in a classical orthogonal polynomial to high order. In the guided spectral transform Lanczos method (GSTLM),[46] f is expanded to lower order, thus not being accurately reproduced but only being a guide for the transformation. This makes the action on the wave function faster. The eigenvalues of the Hamiltonian operator are, however, still obtained accurately. The GSTLM can achieve a CPU time scaling between NlnN and N^{2} and is suitable for massive parallel computing unlike most direct diagonalization methods.[18, 70] It has also proven both efficient and stable in applications to several reactions.[6, 45, 46, 70]
In Figure 2, experimental and computed differential cross-sections are shown for the Cl+CHD3→HCl+CD3 reaction.[73] The experimental results were obtained using molecular beams and the computed values were obtained using the reduced dimensionality rotating line umbrella (RLU) model, employing the GSTLM algorithm just described. In the RLU model,[47] three dimensions are treated explicitly quantum dynamically, viz. the two reactive bonds and the umbrella motion of the three nonreacting hydrogen atoms. A potential energy surface that was tailored for RLU calculations on the Cl+CH4→HCl+CH3 reaction[71] was adapted to the Cl+CHD3→HCl+CD3 reaction. The changes made account for the fact that the zero-point energies are different for the two isotopic versions of the reaction and these were included in the potential energy surface for the modes not treated quantum dynamically.
Figure 2 illustrates that as the scattering energy is increased, the differential cross-sections move from backward to sideways scattering. This can be understood from the fact that for small collision energies only head-on or nearly head-on collisions lead to reaction, yielding backward scattering of the products. As the collision energy increases, a larger centrifugal barrier can be overcome and thus collisions with larger impact parameters can lead to reaction. The impact parameter is a classical measure of by how much the centers of mass of the collision partners would miss each other if the interaction potential was zero. The larger the impact parameter, the more feasible that the products can move forward. A larger impact parameter corresponds to a larger total angular momentum quantum mechanically. The overall trends are the same for the experimentally measured and the computed differential cross-sections. It is seen that the RLU reduced dimensionality scattering calculations do not reproduce the experimental results in detail, but they give the correct overall trends.
It is possible to carry out full-dimensional CC calculations for some reactions involving four-atoms.[36, 37, 79] Still, four-atom reactions are mostly treated with reduced dimensionality approaches.[45, 46, 74, 80-84]
Time-Dependent Quantum Dynamics
Time-dependent quantum dynamics has been reviewed on several occasions, see for instance Refs. [17, 18, 85, 96] and references therein. Solving the TDSE results in an initial value problem, which often is solved by propagating a wave packet in time. In this section, I give a basic description of the TDWP approach in one dimension. I then turn to the MCTDH method, which aims to treat high-dimensional quantum dynamics. I also briefly describe Feynman path integrals and the related RPMD method, which can be used to obtain approximate but quite accurate thermal rate constants for large systems.
The TDWP approach
In early quantum dynamics calculations, the CC approach was more widely used than the TDWP methods. During the last 20 years or so, the TDWP approach has however, become more popular, especially, for polyatomic molecules. This is largely due to the incorporation of efficient and accurate methods like the fast Fourier transform (FFT) and the Chebyshev expansion for wave packet propagation, and development of new approaches[17, 87, 88] such as the flux formalism, the one column S-matrix technique, the reactant/product decoupling method, and the MCTDH method.[89-93]
Perhaps the most important feature of the TDWP method is that it requires only Hamiltonian-vector actions on the wave function,[87, 88, 94] thus no matrix diagonalization. As a result, little core memory is needed and the algorithms often show a nearly linear scaling ( NlnN) as the basis size N grows. TDWP algorithms are also well suited for parallel computing and they give results for several energies in a single calculation, although only for one initial state. The CC approach conversely gives results for all state-to-state transitions in the same calculation, but only for a single energy. Efficient wave packet methods for state-to-state reactive scattering calculations on AB + CD → ABC + D reactions have been developed, see for example, Refs. [95-97].
A wave packet can be obtained as a superposition of eigenfunctions of the Hamiltonian. It is, therefore, not a stationary state and it has a finite extent. For a bimolecular chemical reaction, the wave packet is usually initialized in the asymptotic region, traveling toward the interaction region. As the wave packet is localized in coordinate space, it follows from the uncertainty principle that it has a spread in momentum space. This means that if the average momentum is zero the wave packet will spread along the spatial coordinate in both the positive and the negative direction. This also implies that for low average kinetic energy of the wave packet, it may have components traveling toward the grid edge rather than toward its center. This, and the long propagation time required for the low energy components, makes the wave packet approach more favorable at higher kinetic energies.
For problems requiring calculations at low collision energy, time-independent approaches are usually favored over time-dependent approaches. Photodissociation studies conversely are ideal for TDWP calculations as they often start from a single initial state, the vibrational ground state on the lower potential energy surface, and then dissociate on a repulsive upper surface yielding large kinetic energies.
The TDWP approach has some similarities with the classical trajectory approach. A classical trajectory calculation consists of three main parts, viz.
choosing initial conditions,
propagating forward in time,
analyzing the outcome.
This structure is the same in the TDWP approach, but there are also differences. In classical mechanics, each atom is moved in a continuous space according to its velocity and a time step that has been chosen. In the TDWP approach, a time step is also chosen, but here the propagation involves updating wave function amplitudes at chosen discrete grid points rather than moving individual atoms. The wave packet is thus represented by amplitudes at grid points.
The changes in amplitudes with time are determined by the action of the time evolution operator Û(t)=exp{−iĤΔt/ℏ} on the wave packet. This involves the action of the Hamiltonian Ĥ on the wave packet. In particular, the action of the kinetic energy operator can be time consuming to perform and two common techniques to do this efficiently will be described, viz. the FFT and the discrete variable representation (DVR) methods. There are several DVR methods and which to choose depends on the shape of the potential and this determines where the grid points will be placed. This is thus part of setting up the initial conditions and is described under that heading. Thereafter the popular split-operator propagator (SOP) is described. I end this section on the TDWP approach with a brief description of how the asymptotic analysis can be performed.
Initial conditions
The initial form of the wave packet is often chosen to be of Gaussian shape in the scattering coordinate
Ψ(x,t=0)=12πσ24e−ikox−(x−xo)2/4σ2,(21)
with ℏko being the average momentum of the wave packet and σ relating to its width. The initial wave packet is placed on the chosen grid. For multidimensional problems, the initial Gaussian is often multiplied by eigenfunctions of the reacting molecule(s).
The FFT
The kinetic energy operator in the Hamiltonian contains one or more second derivatives, which are to act on the wave packet. In the Fourier transform method, this is done by first Fourier transforming the wavefunction Ψ(x) to the momentum representation and thereafter multiplying each momentum component by −k2, where k is the wave number.[98-100] Finally, the resulting function is transformed back to the original coordinate representation. In one-dimension, the Fourier transform of Ψ(x) can be written
Ψ(k)=12π∫−∞+∞Ψ(x)e−ikxdx.(22)
The reverse transform is then
Ψ(x)=12π∫−∞+∞Ψ(k)eikxdk,(23)
whereby
d2Ψ(x)dx2=−12π∫−∞+∞k2Ψ(k)eikxdk.(24)
As mentioned, the wave packet is discretized on a grid. Therefore, the integrals in Eqs. (22)-(24) can be replaced by summations, resulting in a discrete Fourier transform. The FFT method is a numerically efficient implementation of the discrete Fourier transform. The grid has to be large enough that the wave packet is zero at the boundaries.[101]
The FFT scales as favorably as NlnN with the number of grid points N and is therefore often the choice when many grid points are required. If symmetry exists, it can be used to reduce the computational effort.
The discrete variable representation
Here I describe the DVR[102, 103] for evaluating the action of the kinetic energy operator on the wave packet. The principle is to take a set of orthonormal basis functions and linearly recombine these to form new orthonormal functions. This is done such that each of the new functions, the DVR functions, are oscillatory but has large amplitude at only one of the grid points and is zero at all other grid points. Mathematically we can describe this as follows.
The wavefunction Ψ(x,t) is expanded in an orthonormal basis ϕi(x):
Ψ(x,t)=∑iNai(t)ϕi(x)(25)
where
ai(t)=∫ϕi*(x)Ψ(x,t)dx.(26)
The integral can be approximated by an N-point quadrature as
ai(t)=∑jNωjϕi*(xj)Ψ(xj,t),(27)
where ωj is the relevant quadrature weight at grid point x_{j}. Equation (27) can be inserted into Eq. (25) to give
Ψ(x,t)=∑iN∑jNωjϕi*(xj)Ψ(xj,t)ϕi(x).(28)
We can define the DVR functions by
ψj(x)=ωj∑iNϕi*(xj)ϕi(x),(29)
where the weight factor ωj is chosen so that the ψj's become orthonormal. It follows that
Ψ(x,t)=∑jNΨjψj(x),(30)
with
Ψj=ωjΨ(xj,t)(31)
being the expansion coefficient of the jth DVR basis function ψj(x).
Finally, we can write the second derivative of Ψ(x,t) as
These derivatives are normally easy to obtain. They need only be calculated once for each grid point and can then be stored and used throughout the propagation.
There are several different DVR grids to choose between as these can be generated as the N Gausssian quadrature points for any classical orthogonal polynomial of order N.[102] The choice of polynomial also determines the weights to be used.
As a specific illustration, set the basis functions to be
and use them to form DVR functions according to Eq. (29). In Eq. (34), i={0,1,2,…,N−1} where N is the number of basis functions chosen. The grid points (pivots) will be xj=xmin+(j+1/2)Δx with j={0,1,…,N−1} and Δx=(xmax−xmin)/N. The weight functions are ωj=Δx. Figure 3 illustrates one of the nine resulting DVR functions, obtained from Eq. (29) with ϕi(x) as in Eq. (34), for the choice xmin=0, xmax=1, and N = 9. The particular DVR function in the figure is for i = 5. Note that the DVR function is nonzero at only one of the grid point. This also holds for the other DVR functions and they are all nonzero at different grid points. This is numerically convenient when performing the Hamiltonian action on the wave function in the DVR representation.
Propagation
For a time-independent Hamiltonian, the solution to the TDSE may be expressed
Ψ(t+Δt)=e−iĤΔt/ℏΨ(t).(35)
Thus, the action of the time evolution operator exp{−iĤΔt/ℏ} on the wave function Ψ(t) has to be evaluated in the propagation. Several methods to do this have been developed,[104] like the split-operator,[105] the Lanczos, and the Chebyshev[106, 107] methods. I will only describe the second-order split-operator method. Additional methods are described in Refs. [17, 18] and references therein.
The split-operator propagator splits the kinetic energy operator T̂ and the potential energy operator V̂ into different exponentials in the time evolution operator. As T̂ and V̂ do not commute, this is an approximation. Accepting this approximation we can write
e−iΔtĤℏ≈e−iΔtT̂2ℏe−iΔtV̂ℏe−iΔtT̂2ℏ+O((Δt)3),(36)
where Δt is a time-step.
By making Δt suitably small, the propagation
Ψ(t+Δt)=e−iΔtT̂2ℏe−iΔtV̂ℏe−iΔtT̂2ℏΨ(t)(37)
becomes accurate. The SOP is unitary and therefore numerically stable as it conserves the norm of the wave function.
To carry out the action in Eq. (37), a Fourier transformation of the wave function to momentum space is first performed. The wave function is then multiplied by exp{−ik2Δtℏ/4μ}, where µ is a reduced mass. Thereafter, the wave function is transformed back to coordinate space and multiplied by exp{−iΔtV/ℏ}. Finally another Fourier transform to momentum space, multiplication by exp{−ik2Δtℏ/4μ} and back transformation are carried out. In this procedure, the action of both T̂ and V̂ takes place in their respective local representations.
The splitting of T̂ and V̂ in Eq. (36) so that the potential energy operator is sandwiched between kinetic energy operators is referred to as the potential referenced SOP. Alternatively the kinetic energy operator is sandwiched between potential energy operators, which is referred to as the potential referenced SOP.
Asymptotic analysis of the wave packet
To save computational effort, the grid is chosen as small as possible. If the reactant and/or product quantum states are to be resolved, the corresponding asymptotic regions must be covered. The wave packet should not be allowed to reach the end of the grid as this would be equivalent to reaching an infinite potential wall, resulting in unphysical reflection. Therefore, it is often necessary to absorb the wave packet before it reaches the grid boundary. This can for instance be achieved with the transmission free absorbing potential of Manolopoulos.[108] The analysis of the wave packet is done just before it reaches the absorption region, which is placed as close to the grid edge as possible.
A simple way to obtain reaction probabilities[109, 110] is to record the amplitude of the wave function as a function of time at a chosen (asymptotic) grid point, Ψ(xp,t). By performing a Fourier transform we can switch to the energy domain;
aE,out=12π∫Ψ(xp,t)exp(iEt/ℏ)dt.(38)
Reaction probabilities are obtained as
P(E)=Fout/Fin,(39)
where the outgoing flux is
Fout=koutμ|aE,out|2.(40)
Here μ is the reduced mass in the product scattering coordinate and k_{out} is the magnitude of the outgoing wave vector. We have the incoming flux
Fin=kinμin|aE,in|2,(41)
with
|aE,in|2=(μinℏkin)2|akin|2.(42)
If the reactants are initiated at large enough separation that the interaction potential vanishes, |akin|2 is for a Gaussian wave packet given by[109]
|akin|2=2πσexp[−2σ2(kin−ko)2],(43)
In several dimensions, unless state-resolved quantities are desired, Ψ(xp,t) is first obtained as
Ψ(xp,t)=∫ϕ*(r)Ψ(xp,r,t)dr,(44)
where r stands for all degrees of freedom except the scattering coordinate, and then the above procedure is applied.
The MCTDH method
The MCTDH approach has been extensively reviewed, see for instance Refs. [85, 111, 112]. Here I only give a short overview with special attention to the calculation of thermal rate constants.
The MCTDH approach differs from the standard TDWP approach in that it uses time-dependent basis functions. This results in the possibility of using fewer basis functions, which from a numerical point of view becomes very important in high dimensionality. It however, also leads to more complicated equations of motion, suggesting that it is in high dimensionality that MCTDH is most useful.
where φjκ(κ)(xκ,t) is a time-dependent basis function, called a single-particle function, and Bj1…jf is an expansion coefficient. In this expression, there are f degrees of freedom, each of which is expanded in n_{i} single particle functions.
The equations of motion for the expansion coefficients Bj1…jf and the single-particle functions φjκ(κ)(xκ,t) can be derived from the Dirac-Frenkel variational principle[91, 113]
〈δΨ|i∂∂t−Ĥ|Ψ〉=0.(46)
Each of the single-particle functions is represented in a time-independent (DVR or FFT) basis set { χiκ(κ)(xκ),iκ=1,2,..,Nκ} in that coordinate:
φjκ(κ)(xκ,t)=∑iκ=1Nκcjκ,iκ(κ)(t)χiκ(κ)(xκ).(47)
The initial form of the single particle function is often a Gaussian.
For high-dimensional problems, the major numerical effort comes from propagating the B-coefficients and the effort scales as nf+1, if all ni=n.[113] If ni=Ni, single particle function i becomes time-independent. If ni=Ni for all i, then Eq. (45) becomes equivalent to the standard TDWP approach, which scales as Nf+1 if all Ni=N. Often n_{i} can be substantially smaller than N_{i}. Thus substantial computer time may be saved using MCTDH rather than the TDWP approach for high-dimensional problems.
In calculating thermal rate constants, a very convenient expression is given in Eq. (9). It requires the cumulative reaction probability at enough energies to evaluate the integral, but not state-to-state reaction probabilities. To obtain the cumulative reaction probability, it is sufficient to know whether products or reactants are formed, not in which quantum state they form. Therefore, only a region near the barrier needs to be treated in the calculations.
The probability of reaction is related to the flux into products, which is given by the flux operator F̂ in Eq. (15). This operator is however, singular so for numerical reasons a thermalized flux operator,
F̂T=e−Ĥ2kBTF̂e−Ĥ2kBT(48)
is used. The eigenstates |fT〉 of this thermal flux operator can be obtained at the temperature T by diagonalizing the operator by an iterative Lanczos scheme adapted to work well with the MCTDH wave functions. The cumulative reaction probability for J = 0 can be expressed in terms of the thermal flux eigenstates as[114]
Here f_{T} and fT′ are the eigenvalues of |fT> and |fT′>, respectively. By propagating the thermal flux eigenstates |fT′> in Eq. (49) by the time evolution operator, NJ=0(E) can be obtained. NJ=0(E) can be evaluated at a range of energies by simply changing E in Eq. (49). The thermal rate constant is then obtained by inserting NJ=0(E) in Eq. (10).
The Feynman path integral approach and RPMD
In this section, I review the Feynman path integral approach and then with this background I discuss the RPMD approach to quantum dynamics.
The Feynman path integral approach
The Feynman path integral approach to quantum mechanics[115] 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 the Feynman path approach, all possible paths are considered that connect an initial point with a final point at a certain later time. Each path has a phase that depends on the classical action associated with that particular path. All paths are summed up, accounting for their phases, which results in a quantum mechanical amplitude, K(b,tb;a,ta), the square of which gives the probability of moving from a to b in time t=tb−ta. K(b,tb;a,ta) is called the kernel or the propagator and the path integral expression for it in one-dimension may be written[115]:
K(b,tb;a,ta)=∫abeiS[b,tb;a,ta]/ℏDx(t),(50)
where the script D signifies that all paths are to be included. The classical action may be written
S[b,tb;a,ta]=∫tatbL(ẋ,x,t)dt,(51)
where L is the Lagrangian and the dot on x signifies time derivative.
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 ℏ→0, 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 ℏ 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
The first term on the right-hand side vanishes since δx=0 at the end points. We thus have the condition
ddt(∂L∂ẋ)−∂L∂x=0,(55)
which is the classical Lagrangian equation of motion. Thus, in the classical limit, obtained by letting ℏ→0, only paths obeying classical mechanics survive. We have thus seen how classical mechanics is naturally embedded in the path integral approach.
In reality ℏ 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
Û(t)=e−iĤt/ℏ(56)
the propagator can be written
K(x,x′;t)=〈x|Û(t)|x′〉(57)
in the position eigenstate basis. We now split Û(t) into N equal time slices and write
Û(t)=∏i=1NÛ(t/N).(58)
By inserting the resolution of identity
I=∫dx|x〉〈x|,(59)
N + 1 times in the time evolution operator we get
Û(t)=∫dx1∫dx2…∫dxN+1|xN+1〉〈xN+1|Û(t/N)|xN〉(60)
×〈xN|Û(t/N)|xN−1〉…〈x2|Û(t/N)|x1〉〈x1|.(61)
Next split Ĥ=T̂+V̂ in the time evolution operator so that T̂ and V̂ appear in different exponentials in each matrix element 〈xj+1|Û(t/N)|xj〉. This is an approximation since the kinetic and potential operators do not commute. The approximation, however, becomes more accurate as the time step t/N becomes smaller. Thus if a sufficient number of time slices are used, good accuracy can be achieved. Focusing on the kinetic part of the matrix element, that is, 〈xj+1|exp[−ip̂2t/(2mℏ)]|xj〉, and inserting
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.
We shall now see how the average value of an operator can be evaluated by imaginary time path integrals. Consider the partition function
Z=tr{exp(−βĤ)}=〈x|exp(−βĤ)|x〉(68)
where β=1/(kBT). By setting it/ℏ=β and xN+1=x1=x in Eq. (66), we obtain
Integrating out the momentum variables in Eq. (69) yields
Z=(mN2πℏ2β)N2∫dx2…∫dxNe−βVeff,(70)
where
Veff=∑j=1N[mN2ℏ2β2(xj+1−xj)2+V(xj)N].(71)
As the coordinates {x1,x2,…,xN} 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 V_{eff} 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 Â can be obtained as
The multidimensional integrals in Eq. (72) can be evaluated by Monte Carlo sampling[116] 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, 〈Â〉 converges to the exact value.
An alternative to evaluating 〈Â〉 by PIMC is to use the path integral molecular dynamics (PIMD) method. PIMD was introduced by Parrinello and Rahman[119] 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[120] 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.[121] and Nosé–Hoover thermostats.[122, 123]
Ring polymer molecular dynamics
Many dynamical quantities can be calculated from real-time correlation functions.[25] The standard form for a real-time correlation function is
cAB(t)=1Ztr[e−βĤÂ(0)B̂(t)],(74)
where
B̂(t)=e+iĤt/ℏB̂e−iĤt/ℏ(75)
and Ĥ is the ordinary Hamiltonian operator. An alternative form is the Kubo-transformed correlation function
c∼AB(t)=1βZ∫0βdλtr[e−(β−λ)ĤÂ(0)e−λĤB̂(t)],(76)
which has more classical-like properties than the standard form in Eq. (74). The Fourier transform of the Kubo-transformed correlation function,
C∼AB(ω)=∫−∞∞e−iωtc∼AB(t)dt,(77)
is related to the Fourier transform of the standard correlation function,
CAB(ω)=∫−∞∞e−iωtcAB(t)dt,(78)
by
CAB(ω)=βℏω1−e−βℏωC∼AB(ω),(79)
so that either one can be obtained form the other and they are identical for ω=0. Diffusion coefficients for example can be calculated as
D(T)=13∫0∞cvv(t)dt=13∫0∞c∼vv(t)dt,(80)
where cvv(t) is the velocity–velocity autocorrelation function and c∼vv(t) 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[24] 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[127] 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).[128] 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 Â or B̂ 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.
RPMD accounts for zero point energy and tunneling, but there are also limitations of RPMD. Interference is not handled and RPMD is limited to calculating observables obtained from Kubo-transformed correlation functions. Also, the operators Â and B̂ should be configurational and one or both of them should be linear in x. The calculation of thermal rate constants should thus not involve the flux operator, found in cff(t) [Eq. (13)] and cfs(t) [Eq. (11)], nor the Heaviside step function, found in cfs(t) and css(t) [Eq. (11)], which is highly nonlinear. Still, Craig and Manolopoulos have shown that RPMD can be implemented to work well for thermal rate constants.[124] We note also that momentum dependent operators, like the flux operator, can be rewritten in terms of a time derivative of position.
In the calculation of thermal rate constants, the Kubo-transformed flux-side correlation function
x¯ and p¯ define the ring polymer centroid in coordinate and momentum space, respectively. Obtaining Eq. (86) involves a few steps.[124] The physical meaning of the equation is that the δ-function pins the initial centroid position to the transition state, placed at x‡, whereas p¯0/m represents the initial centroid flux passing the barrier and the Heaviside step function h_{s} is used to check whether products are formed or not.
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.[130] 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 H+CH4→H2+CH3 reaction are shown. Results from MCTDH[7] and RPMD[126] calculations are shown as well as results from canonical variational transition state theory with microcanonically optimized multidimensional tunneling[7, 8] and classical transition state theory (TST).[126] The potential energy surface that has been used to calculate all rate constants in the table is due to Espinosa-Garcia.[8] 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.
Table 4. Calculated thermal rate constants for the H+CH4→H2+CH3 reaction on the PJEG potential energy surface[8] obtained using MCTDH,[7], RPMD,[126], canonical variational transition state theory with microcanonically optimized multi-dimensional tunneling[7, 8] and classical transition state theory.[126]
T/K
kMCTDH
kRPMD
kCVT/μOMT
kTST
The rate constants are given in cm^{3}/mol/s and the numbers in parenthesis denote powers of 10.
2000
–
1.49(−11)
1.70(−11)
1.51(−11)
1000
–
3.31(−13)
3.58(−13)
2.14(−13)
800
–
6.28(−14)
6.72(−14)
3.15(−14)
600
–
4.60(−15)
5.00(−15)
1.42(−15)
500
–
6.51(−16)
6.96(−16)
1.25(−16)
400
2.81(−17)
3.93(−17)
4.17(−17)
3.59(−18)
300
2.81(−19)
4.73(−19)
5.06(−19)
8.08(−21)
225
1.01(−21)
1.94(−21)
2.28(−21)
5.37(−24)
200
–
1.63(−22)
1.87(−22)
1.24(−25)
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.
Summary and Concluding Remarks
I have summarized some basic aspects of the time-independent CC method, the TDWP approach including the MCTDH method, and the RPMD method. In exact calculations results obtained with time-dependent and time-independent quantum dynamics agree. Dependent on the particular problem at hand, one of the approaches may, however, be numerically more convenient.
In typical implementations of the TDWP approach, each run gives results for reactant(s) in a single rovibrational state, but for a range of relative translational energies. Very low translational energies can cause numerical problems. Therefore, TDWP methods work well in for instance photodissociation problems where often only the vibrational ground state is of interest on the lower potential energy surface and the upper surface is commonly repulsive so that products form with substantial kinetic energy. The CC approach conversely gives transition probabilities between all initial and final states at a given total energy in each run. It also works well at low translational energies and therefore has advantages for low temperature applications.
A major obstacle in quantum dynamics is the exponential scaling of the numerical effort with basis size. The main advantage of the MCTDH approach compared to the standard TDWP approach is the gain in CPU time. This comes from the usually much smaller prefactor in the MCTDH numerical effort, whereas otherwise both approaches scales exponentially. For chemical reactions, the MCTDH approach is therefore typically limited to 12-dimensions or less. As the MCTDH wave function is expressed in single particle functions, which in turn are expressed in time-independent basis functions, the MCTDH wave function is viewed as a two-layered wave function. To gain even more speed, there is a multilayer version, ML-MCTDH, of MCTDH,[133, 134] which uses further layering. There is also G-MCTDH.[135] In G-MCTDH, some degrees of freedom are treated with parametrized Gaussians whereby larger systems can be treated, but accuracy can be lost.
The RPMD method is based on the path integral approach to quantum mechanics. The idea is to propagate a ring polymer, that is, a set of beads connected by harmonic springs, just as in PIMD, but with the bead masses being the physical masses of the atoms they represent and focusing on calculating Kubo-transformed correlation functions. This means that classical dynamics is performed but in an extended phase space. RPMD is approximate but has been successfully applied to a large variety of problems.[25, 124, 125, 136-139] Rate constants are typically obtained to within a factor of two or three. It will be most interesting to see how RPMD will be used in the future.
Acknowledgment
The author is grateful to Jens Poulsen, Magnus Gustafsson, and Sture Nordholm for their useful discussions.