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The second-order Ehrenfest method

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Abstract

This article describes the Ehrenfest method and our second-order implementation (with approximate gradient and Hessian) within a CASSCF formalism. We demonstrate that the second-order implementation with the predictor–corrector integration method improves the accuracy of the simulation significantly in terms of energy conservation. Although the method is general and can be used to study any coupled electron–nuclear dynamics, we apply it to investigate charge migration upon ionization of small organic molecules, focusing on benzene cation. Using this approach, we can study the evolution of a non-stationary electronic wavefunction for fixed atomic nuclei, and where the nuclei are allowed to move, to investigate the interplay between them for the first time. Analysis methods for the interpretation of the electronic and nuclear dynamics are suggested: we monitor the electronic dynamics by calculating the spin density of the system as a function of time.

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Acknowledgments

This work was supported by UK-EPSRC Grant EP/I032517/1. All calculations were run using the Imperial College High Performance Computing service. The original SA CP-MCSCF programs were written by Thom Vreven. The work on the Ehrenfest programs was initiated by Patricia Hunt (see the supplementary information of reference [9]).

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Correspondence to Michael A. Robb.

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Dedicated to the memory of Professor Isaiah Shavitt and published as part of the special collection of articles celebrating his many contributions.

Appendix: Computation of the energy and the gradient of a complex wavefunction

Appendix: Computation of the energy and the gradient of a complex wavefunction

1.1 Energy computation

Let us consider the expansion of the TD wavefunction in the eigenstate basis set as defined in Eq. (10) with complex coefficients \(\{c_k\}\). \(\fancyscript{H}_e\) is the electronic Hamiltonian operator and \({{\varvec{H}}}\) its matrix representation in the eigenstate basis with elements \({{\varvec{H}}}_{kl}=\langle \tilde{\phi }_k | \fancyscript{H}_e | \tilde{\phi }_l \rangle\). The energy of the TD wavefunction is computed as the expectation value of this operator:

$$\begin{aligned} E=\langle \varPsi | \fancyscript{H}_e | \varPsi \rangle = \sum _k {\sum _l{ c^*_kc_l {{\varvec{H}}}_{kl}}} \end{aligned}$$
(30)

Note that in the eigenstate basis, \(\mathbf{H}_{kl}=0\) for \(k \ne l\) so the double sum reduces to one. The energy expression then reads:

$$\begin{aligned} E= \sum _k {|c_k|^2 E_{k}} \end{aligned}$$
(31)

Here, we see that the energy depends only on the weight of each eigenstate and not on their relative phase. From an implementation point of view, instead of repeating the operations for the real and imaginary components, we create a real wavefunction that has the same energy and using directly the machinery already programmed. For that, one needs to rotate all the complex coefficients in the TD vector expansion so that they are all real but conserving their magnitude. So the energy evaluated with the vector rotated to real is equal to the energy of the complex TD vector. Note this is only true because the rotation is done in the eigenstate basis. In general, we can compute the expectation value of an operator with the wavefunction rotated to real only if the operator is diagonal in the basis set we do the rotation in.

1.2 Gradient computation

The Hellmann–Feynman term of the gradient is defined as the partial derivative of the energy with respect to a nuclear distortion \(R_I\). To consider the intrinsic dependence of the energy, we assume an expansion in exact eigenstates. By applying the product rule, we obtain:

$$\begin{aligned} E^{R_I}= \sum _k {\nabla _{R_I}(|c_k|^2) \cdot E_{k}} + \sum _k {|c_k|^2 \cdot \nabla _{R_I}(E_{k})} \end{aligned}$$
(32)

The second term on the right hand side is the average of the gradient of each electronic eigenstate weighted by their occupation. It represents the change in potential energy staying on the same potential, i.e. keeping the same occupation on each electronic eigenstate. The first term however is the change in potential energy due to change in occupation of the electronic eigenstates because of non-adiabatic transitions. To calculate the derivative of \(|c_k|^2\) with respect to a nuclear distortion, we can invoke the time derivative by applying the chain rule:

$$\begin{aligned} E^{R_I}= \sum _k {\frac{\partial |c_k|^2}{\partial t} \frac{1}{\dot{R_I}} \cdot E_{k}} + \sum _k {|c_k|^2 \cdot \nabla _{R_I}(E_{k})} \end{aligned}$$
(33)

The time derivative of the norm squared of the expansion coefficient \(|c_k|^2\) with respect to time can be obtained using Eq. (11):

$$\begin{aligned} \frac{\partial |c_k(t)|^2}{\partial t} = - \sum _{l,J}{\left( c_l(t)c_k^*(t)+c_l^*(t)c_k(t)\right) \cdot d_{kl}^J(R) \cdot \dot{R_J} } \end{aligned}$$
(34)

By inserting this in Eq. (33), it reads:

$$E^{R_I}= - \sum _{k\ne l} { \left( c_l(t)c_k^*(t)+c_l^*(t)c_k(t)\right) \cdot d_{kl}^J(R) \cdot E_{k}} + \sum _k {|c_k|^2 \cdot \nabla _{R_I}(E_{k})}$$
(35)

We see that the non-adiabatic coupling \(d_{kl}^J(R)\) is present in the term representing the change in energy due to electronic transitions. Using the relation \(d_{kl}^J(R)=-d_{lk}^J(R)\) gives:

$$E^{R_I}= - \sum _{k\ne l} { c_l^*(t)c_k(t) \cdot d_{kl}^J(R) \cdot \left( E_{k}-E_{l}\right) } + \sum _k {|c_k|^2 \cdot \nabla _{R_I}(E_{k})}$$
(36)

Using Eq. (18), it becomes:

$$\begin{aligned} E^{R_I}& = \sum _{k \ne l} { c_l^*(t)c_k(t) \cdot \langle \tilde{\phi }_l | \nabla _{R_I} (\fancyscript{H}_{e}) | \tilde{\phi }_k \rangle } \\ & \quad + \sum _k {|c_k|^2 \cdot \langle \tilde{\phi }_k | \nabla _{R_I} (\fancyscript{H}_{e}) | \tilde{\phi }_k \rangle } \end{aligned}$$
(37)

The Hellmann–Feynman term of the gradient is the expectation value of the derivative of the Hamiltonian operator. On one hand, the diagonal terms represent the weighted average potential and they depend only on the norms of the expansion coefficients. On the other hand, the off-diagonal terms represent the change in energy due to non-adiabatic transitions and they do depend on the relative complex phase of the expansion coefficients. For this reason, one can not construct a real wavefunction whose gradient would be equal to the gradient of a complex wavefunction.

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Vacher, M., Mendive-Tapia, D., Bearpark, M.J. et al. The second-order Ehrenfest method. Theor Chem Acc 133, 1505 (2014). https://doi.org/10.1007/s00214-014-1505-6

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