P−) and reactive (P+) transition probabilities on the adiabatic ground state as functions of time with all calculations performed in adiabatic representation. DVR approach as a function of time for the following cases: (a) Four-mode model (system); (b) System associated with five bathing modes; (c) System associated with twenty bathing modes.
Overview of methods
For dissociative and reaction coordinates it is convenient to introduce discrete representation of the wave function, namely grid methodology. From the computational point of view, some of the methods need much less information than, for example, state- and energy-resolved differential cross sections.
Classical dynamics
Classical equation of motion
2mr2 +ν(r) (1.2.5) where j is the spin quantity, ν(r) is the intramolecular potential and m is the reduced mass of the diatomic molecule, m=mBmC/(mB+mC). If a reaction occurs, it is convenient to describe the system using the r′R′ coordinates shown in Fig.
Semiclassical theories
Feynman path integrals
The amplitude for a final position, that is, ψ(x2, t2) can be obtained by summing or integrating over all such "transition" amplitudes multiplied by the amplitude for a given initial position. We now assume that the propagator can be written as an integral over all "paths" leading from (x1, t1) to (x2, t2) each weighted by a phase factor.
Short-time propagator
In some cases, this integration can be performed analytically, namely if the force is at most linear in the x-coordinate, and in the general case for a quadratic Hamiltonian.
Evaluation of kinetic and potential energy operators and Wave packet
Fourier Transform Method
DVR method
A complex FFT routine gives a certain order of the frequencies starting from zero and ending with −1/N△x as output, i.e. we have [7]. The evaluation of the derivative at each of the N grid points requires N operations and therefore the DVR method is an N2 method.
Time propagation
S†T†φ(t) (1.4.47) where D is the diagonal matrix (L ×L) containing the eigenvalues of the tridiagonal matrix; This is an (N×L) matrix with recursion vectors of length N (the number of grid points ); and Is an (L×L) matrix containing the eigenvectors of the tridiagonal matrix. The number of Lanczos recursions depends on the length of the time step and the size of the Hamiltonian.
Gaussian Wave Packet based methods
The classical path equations
We thus see that the classical path equations perform the simultaneous integration of the "classical" equations of motion (Eq.
Corrections to the classical path equations: Hermite corrections 37
The method corrects both in the SCF and MCSCF direction, and is therefore ideal for correcting the classical path equations. The first equation introduces the classical path picture (the center of the wave packet follows a classical equation of motion) and the two other equations ensure that the.
Time-dependent discrete variable representation method (TD-
In fact, the derivation has the possibility of forming a rigorous expression for Vef f′ and Vef f′′ using the Dirac-Frenckel variational principle. TDDVR can be applied to any potential by means of a force that Newtonian mechanics does not really know.
Introduction
Finally, moving grid points (TDDVR) significantly reduce computation costs compared to fixed grid points (DVR). In this respect, TDDVR is similar to the distributed approximation function (DAF) approach [50, 51], which again differs from TDDVR in the sense that TDDVR lattice points propagate according to the “classical” dynamics of the system.
Formulation of the TDDVR method for one - dimensional multi -
2.2.1)) and due to the particular choice of the form of the TDDVR basis functions as defined in Eq. Therefore the matrix Z cannot facilitate the convergence of the solution of the quantum equation of motion as Y if.
Formulation of the TDDVR method for multi - dimensional multi -
The matrix element of the wave function in Eq. 2.3.4) is the wave function for the lth surface and we can expand the time-dependent wave function (Ψl({sk}, t)) as follows,. 2.3.1)) and due to the particular choice of the form of the TDDVR basis functions as defined in Eq.
Summary of the TDDVR approach
In the quantum regime, the dynamics of the wavepacket center obeys the Ehrenfest theorem, which facilitates the movement of lattice points. The number of grid points required to achieve convergence to the DVR depends on the initial position of the wave packet.
Tunneling through an Eckart potential
Formulation of equation of motion
Similarly, the initial amplitudes of the TDDVR wave function at the grid points (sis) are obtained from the GWP (Φ(si, t0)) with the same numerical value of the parameters used in the DVR, except zero moment vector (k and also through the first two terms of the Hamiltonian matrix (Eq. Using the properties of the DVR functions, we can write normalized amplitudes as di(t= 0) =A−1/2i,i Φ(xi) (3.1.9) The transmission probability on the reactive side of the barrier can is determined by 2ImAxi⋆ >0, i.e. the sum of the square of the amplitudes of the lattice - points on the reactive side of the barrier gives the probability of tunneling.
Results and discussion
It means when we use a large grid, a sufficient number of grids - the points remain on the reactive side of the barrier for a long time and we get the desired profile. Instead of increasing the grid points, the classical movement can be raised at the time when the tunnel probability reaches the converged value with less number of grid points (≤150) and the same profile can be achieved. 3.4, we compare the tunneling probabilities calculated by quantum - classical (TDDVR) and quantum (DVR) methods with 250 grid points.
Tunneling through a symmetric double well potential
The Hamiltonian and the equation of motion
The center of the wave packet (sc) and its momentum (psc) are considered time-dependent variables, while γ and width (A) are considered time-independent withReγ =ReA = 0. The compact form of the matrix equation and the classical equations of on the way read as follows,. It is natural to expect that one can achieve the exact converged description of the observables with a minimal number of basis functions (DVR) and obtain quantum corrections of the classical variables if the rigorous expression of ˙psc is used.
Results and discussion
This explanation finds support when we monitor the center of the wavepacket, sc as a function of time. In Table 3.2 we present the CPU time (sec) required to propagate the system localized in the minimum of the left well with zero initial momentum (psc = 0) for different initial states (0,0) and (0,2) of the hot bath. QF not only drives the motion of the center of the wavepacket, sc, deep inside.
Introduction
The one dimensional models and TDDVR equations of motion
- Simple avoided crossing
- Dual avoided crossing
- Extended coupling with reflection
- TDDVR equation of motion
- Initialization, propagation and projection
- Results and Discussion
The equation for the movement on the kth curve, in matrix form, is i¯hA ˙Ck = HtkkCk+AX. This model is interesting as well as challenging for total energy below the asymptotic energy of the upper adiabatic potential energy curve (i.e. the center of the wave packet, sc, and its momentum, psc as functions of time, (b) Few DVR networks - points (si) and TDDVR grid - points (si(t)) as functions of time.
![Table 4.1: Potential parameters [146] and other useful data. 1 ˆ ǫ = 100 KJ/mol and 1 τ = 10 −14 s.](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10443812.0/91.918.164.815.279.969/table-potential-parameters-146-and-other-useful-data.webp)
The Born - Oppenheimer treatment, multi - dimensional models and
- The Born - Oppenheimer Treatment
- Multi - dimensional models: The quasi - Jahn - Teller “scat-
- TDDVR equations of motion
- Initialization and projection
- Results and disscusion
4.8(c) shows movements of few grid points (trajectories) around the central trajectory (s1c) as functions of time. 4.8(d) presents the change of min momenta (p1is)(Appendix E) associated with the corresponding grid points (s1is) as functions of time. 4.9(b) displays central trajectories (s1c and s2c) and their associated momenta (ps1c and ps2c) as functions of time.
The effective Hamiltonian and TDDVR theory for the EBO equation 119
TDDVR equations of motion
We present the relevant equations of motion in the simplest yet general way for the current perspective. We derive the TDDVR equations of motion for the ground electronic state of the "JT scattering" model, where the corresponding EBO Hamiltonian is composed of the potential energy, u1(r, R) and the effective operator KE, ˆTn′(r, R) or ˆTn′′(r, R ). Substituting the wave function TDDVR (Eq. 4.4.2)) and Hamiltonian EBO into TDSE, we arrive at the following evolution equation for quantum motion.
Initialization and projection
It is worth mentioning that here we have formulated an alternative fixed width approximation of TDDVR where both quantum as well as "classical" equations of motion are derived from first principles without any approximations, nor do we have a "classical" path does not accept (˙ x = mp) around the center of the wave packet. Finally, in a similar way, we expand the time-dependent wave function in the DVR representation by considering harmonic oscillator eigenfunctions as primitive basis. It is important to note that the last term both in Eq. 4.4.5) and (4.4.7) is the contribution due to vector potential in the TDDVR and DVR equation of motion, respectively.
Results and discussion
We have the dynamics on a single surface EBO equation (formulated from doubly coupled adiabatic electronic states), calculated state-to-state vibrational transition probabilities for the reactive and non-reactive modes using TDDVR and DVR equations of motion and have those results in Tables 4.7 - 4.9 presented. We find that TDDVR achieves excellent quantitative agreement and convincingly fast convergence of calculated transition probabilities at total energies, 1.00 and 1.20 eV. Transition probabilities calculated by TDDVR approach not only follow the correct symmetry but also obtain quantitative agreement with DVR.
Introduction
Quantum dynamics of inelastic scattering
Theoretical Aspects of TDDVR
The compact form of the matrix equation for the quantum equation of motion on the l-th surface can be expressed as follows. The explicit expression of an element of the TDDVR coefficients, di1i2..ip,l is, i¯hd˙i1i2..ip,l = 1. The matrix elements (Xk, Yk, Zk and Ak) are defined in equations. 5.2.5) that the couplings between different electronic states are due to the off-diagonal elements of the potential energy matrix (Vll′) where the kinetic energy matrices, Yk and Zk, the even-lattice-points or basis mode functions kth only on the surface , l.
Initialization, Projection and Propagation
It is important to note that at each time step, diabatic coefficients ({ci1i2.ik.ip,l. t)}) obtained from quantum equations of motion are transformed to adiabatic ones ({ei1i2.ik.ip,l (t) }) and used to evaluate the so-called "quantum correction to classical force".
Model systems, results and discussion
5.1(a) displays the inelastic transition probabilities (P1←0− ) with kinetic energy 4.0 eV as a function of time with increasing number of TDDVR grid points (Nq1) in the reaction coordinate. The profile of time-dependent transition probabilities with 50 grid TDDVR - points has excellent agreement with the DVR results (Nq1 = 250). 5.3(b) shows the central trajectories (s1c and s2c) and their associated moments (ps1c and ps2c) as functions of time.
Introduction
Molecular dynamics with linearly coupled model Hamiltonian
- The Model Hamiltonian
- Theoretical aspects of TDDVR approach
- Numerical details and absorption spectra
- Results and Discussion
The effect of the bath states on the system dynamics has been investigated by including more and more numbers of bath degrees of freedom. 6.1(b)-(c) indicate the sharp effect of bath conditions on the system, ie. the peaks of the autocorrelation function are not only much lower in magnitude, but they also appear with more structure. 6.2(b), we have plotted the population of the diabatic S2 state for the dynamics of the nine-mode model and total model Hamiltonian.
Molecular dynamics with a realistic model Hamiltonian
The model Hamiltonian
The stretching coefficient for the coupling of two electronic states is nonzero if the corresponding oscillation mode (for linear coupling) or the product of both oscillation modes (for bilinear coupling) belongs to B1g symmetry. Similarly, the expansion coefficient leading to intrastate coupling is nonzero if the corresponding oscillation mode (for linear coupling) or the product of oscillation modes (for bilinear and quadratic coupling) has Ag symmetry. The fourth sum (G3) represents a single linear off-diagonal expansion term with symmetry B1g for the interstate coupling between states S1 and S2.
Calculation of the spectra
Furthermore, since the propagation of the wave function and the calculated autocorrelation function is up to a finite time, artifacts known as the Gibbs phenomenon develop in the spectrum. Values specified with an asterisk are adjusted values based on comparison of CIS data and MRCI [151–153]/CASSCF [155] data. Values specified with an asterisk are adjusted values based on comparison of CIS data and MRCI CASSCF data [155].
![Table 6.4: quadratic on diagonal coupling constants (in eV) of the S 1 and S 2 diabatic potentials of pyrazine obtained at the CIS level [149]](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10443812.0/166.1188.145.973.220.818/quadratic-diagonal-coupling-constants-diabatic-potentials-pyrazine-obtained.webp)
The numerical details
On the other hand, this choice plays a crucial role when the mode is considered at the classical limit (one grid point) or close to the classical limit (few grid points), because the dynamics are solely or essentially dictated by classical mechanics. In the present calculations, the choice of classical parameters for the dynamics of one grid point is Qi(t0) = 0 and Pi(t0).
Results and discussion
6.9 we present the effect of an increasing number of bath states on the adiabatic S2 state population dynamics of the subsystem. The entangled absorption spectrum of the S2 states for (c) twelve (τ = 50 fs) and (d) twenty-four (τ = 150 fs) mode models is presented and compared with quantum mechanical profile [149]. The effect of the environment on the subsystem can be monitored by calculating the overlap of time-dependent wave functions of the model Hamiltonian with or without including the bathing states.
Computational and theoretical aspects of TDDVR and the
The theory is based on the expansion of the wave function in the Gauss-Hermite series (G - H). These zeros are the roots (zi) of the Nth Hermitian polynomial and can thus be introduced. A5) in (A1) we obtain the expansion of the wave function with respect to the time-dependent basis on the grid (DVR). Again, the icth grid point (skik) is connected to the icth root (xkik) of the Hermitian polynomial. E1) From the properties of the base TDDVR,.
Overview of Methods and Descriptions in Molecular Dynamics
DVR matrix elements A i,j for N = 8
DVR matrix elements X i,j for N = 8
DVR matrix elements Y i,j for N = 8
DVR matrix elements Z i,j for N = 8
Potential parameters and frequencies. Distances are in ˚ A and energy
Potential energy parameters used for the quasi - Jahn - Teller “scat-
Non - reactive state - to - state transition probabilities when calcula-
Reactive state - to - state transition probabilities when calculations are
Non - reactive state - to - state transition probabilities when calcula-
Reactive state - to - state transition probabilities when calculations are
Convergence(*) profile of vibrational transition probabilities from v =
