Surface hopping

Surface hopping is a mixed quantum-classical technique that incorporates quantum mechanical effects into molecular dynamics simulations. Traditional molecular dynamics assume the Born-Oppenheimer approximation, where the lighter electrons adjust instantaneously to the motion of the nuclei. Though the Born-Oppenheimer approximation is applicable to a wide range of problems, there are several applications, such as photoexcited dynamics, electron transfer, and surface chemistry where this approximation falls apart. Surface hopping partially incorporates the non-adiabatic effects by including excited adiabatic surfaces in the calculations, and allowing for 'hops' between these surfaces, subject to certain criteria.Cite error: A list-defined reference named 'book_2002' is not used in the content (see the help page). Surface hopping is a mixed quantum-classical technique that incorporates quantum mechanical effects into molecular dynamics simulations. Traditional molecular dynamics assume the Born-Oppenheimer approximation, where the lighter electrons adjust instantaneously to the motion of the nuclei. Though the Born-Oppenheimer approximation is applicable to a wide range of problems, there are several applications, such as photoexcited dynamics, electron transfer, and surface chemistry where this approximation falls apart. Surface hopping partially incorporates the non-adiabatic effects by including excited adiabatic surfaces in the calculations, and allowing for 'hops' between these surfaces, subject to certain criteria. Molecular dynamics simulations numerically solve the classical equations of motion. These simulations, though, assume that the forces on the electrons are derived solely by the ground adiabatic surface. Solving the time-dependent Schrödinger equation numerically incorporates all these effects, but is computationally unfeasible when the system has many degrees of freedom. To tackle this issue, one approach is the mean field or Ehrenfest method, where the molecular dynamics is run on the average potential energy surface given by a linear combination of the adiabatic states. This was applied successfully for some applications, but has some important limitations. When the difference between the adiabatic states is large, then the dynamics must be primarily driven by only one surface, and not an average potential. In addition, this method also violates the principle of microscopic reversibility. Surface hopping accounts for these limitations by propagating an ensemble of trajectories, each one of them on a single adiabatic surface at any given time. The trajectories are allowed to 'hop' between various adiabatic states at certain times such that the quantum amplitudes for the adiabatic states follow the time dependent Schrödinger equation. The probability of these hops are dependent on the coupling between the states, and is generally significant only in the regions where the difference between adiabatic energies is small. The formulation described here is in the adiabatic representation for simplicity. It can easily be generalized to a different representation.The coordinates of the system is divided into two categories: quantum ( q {displaystyle mathbf {q} } ) and classical ( R {displaystyle mathbf {R} } ). The Hamiltonian of the quantum degrees of freedom with mass m n {displaystyle m_{n}} is defined as: where V {displaystyle V} describes the potential for the whole system. The eigenvalues of H {displaystyle H} as a function of R {displaystyle mathbf {R} } are called the adiabatic surfaces : ϕ n ( q ; R ) {displaystyle phi _{n}(mathbf {q} ;mathbf {R} )} . Typically, q {displaystyle mathbf {q} } corresponds to the electronic degree of freedom, light atoms such as hydrogen, or high frequency vibrations such as O-H stretch. The forces in the molecular dynamics simulations are derived only from one adiabatic surface, and are given by: where n {displaystyle n} represents the chosen adiabatic surface. The last equation is derived using the Hellmann-Feynman theorem. The brackets show that the integral is done only over the quantum degrees of freedom. Choosing only one adiabatic surface is an excellent approximation if the difference between the adiabatic surfaces is large for energetically accessible regions of R {displaystyle mathbf {R} } . When this is not the case, the effect of the other states become important. This effect is incorporated in the surface hopping algorithm by considering the wavefunction of the quantum degrees of freedom at time t as an expansion in the adiabatic basis: where c n ( t ) {displaystyle c_{n}(t)} are the expansion coefficients. Substituting the above equation into the time dependent Schrödinger equation gives where V j n {displaystyle V_{jn}} and the nonadiabatic coupling vector d j n {displaystyle mathbf {d} _{jn}} are given by The adiabatic surface can switch at any given time t based on how the quantum probabilities | c j ( t ) | 2 {displaystyle |c_{j}(t)|^{2}} are changing with time. The rate of change of | c j ( t ) | 2 {displaystyle |c_{j}(t)|^{2}} is given by: