Electron nuclear dynamics with plane wave basis sets: complete theory and formalism

Abstract

Electron nuclear dynamics (END) is an ab initio quantum dynamics method that adopts a time-dependent, variational, direct, and non-adiabatic approach. The simplest-level (SL) END (SLEND) version employs a classical mechanics description for nuclei and a Thouless single-determinantal wave function for electrons. A higher-level END version, END/Kohn–Sham density functional theory, improves the electron correlation description of SLEND. While both versions can simulate various types of chemical reactions, they have difficulties to simulate scattering/capture of electrons to/from the continuum due to their reliance on localized Slater-type basis functions. To properly describe those processes, we formulate END with plane waves (PWs, END/PW), basis functions able to represent both bound and unbound electrons. As extra benefits, PWs also afford fast algorithms to simulate periodic systems, parametric independence from nuclear positions and momenta, and elimination of basis set linear dependencies and orthogonalization procedures. We obtain the END/PW formalism by extending the Thouless wave function and associated electron density to periodic systems, expressing the energy terms as functionals of the latter entities, and deriving the energy gradients with respect to nuclear and electronic variables. END/PW has a great potential to simulate electron processes in both periodic (crystal) and aperiodic (molecular) systems (the latter in a supercell approach). Following previous END studies, END/PW will be applied to electron scattering processes in proton cancer therapy reactions.

Appendix A: Gradients of the END/PW density matrix \(\varGamma_{ij}^{\left( \varvec{kk} \right)} \left( {z^{*} ,z} \right)\) with respect to the electronic Thouless parameters \(z_{\text{ph}}^{\left( \varvec{k} \right)}\), \(\partial \varGamma_{ij}^{\left( \varvec{kk} \right)} \left( {z^{*} ,z} \right) /\partial z_{\text{ph}}^{\left( k \right)}\)

To obtain the $\partial \varGamma_{ij}^{{\left( \varvec{kk} \right)}} \left( {\varvec{z}^{*} ,\varvec{z}} \right) /\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}}$ gradients, the one-electron density matrix $\varGamma_{ij}^{{\left( \varvec{kk} \right)}} \left( {\varvec{z}^{*} ,\varvec{z}} \right)$ in Eq. (29) is first differentiated with respect to $z_{\text{ph}}^{{\left( \varvec{k} \right)}}$

$$\begin{aligned} \frac{{\partial \varGamma_{ij}^{{\left( {\varvec{kk}} \right)}} \left( {\varvec{z}^{*} ,\varvec{z}} \right)}}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }} & = \mathop \sum \limits_{{l,l^{{\prime }} = 1}}^{{N_{\varvec{k}} , N_{\varvec{k}} }} \frac{{\partial \left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } \\ {z^{{\left( \varvec{k} \right)}} } \\ \end{array} } \right)_{il} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }}\left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{ll^{{\prime }} }} \left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } & {z^{{\left( \varvec{k} \right)\dag }} } \\ \end{array} } \right)_{{l^{{\prime }} j}} \\ & \quad + \,\mathop \sum \limits_{{l,l^{\prime} = 1}}^{{N_{\varvec{k}} , N_{\varvec{k}} }} \left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } \\ {z^{{\left( \varvec{k} \right)}} } \\ \end{array} } \right)_{il} \frac{{\partial \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{ll^{{\prime }} }} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }}\left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } & {z^{{\left( \varvec{k} \right)\dag }} } \\ \end{array} } \right)_{{l^{{\prime }} j}} \\ \end{aligned}$$

(60)

where all the matrices in the above equation are defined just after Eq. (25). In order to obtain an expression appropriate for coding, it is necessary to apply the following matrix relationships [7]:

$$\frac{{\partial A_{bc} }}{{\partial A_{de} }} = \delta_{bd} \delta_{ec} ;\quad \frac{{\partial X\left( x \right)^{ - 1} }}{\partial x} = - X\left( x \right)^{ - 1} \frac{\partial X\left( x \right)}{\partial x}X\left( x \right)^{ - 1}$$

(61)

where $A$ and $X\left( x \right)$ are matrices and $x$ is a variable. By applying these relationships to the derivative matrix terms in the first sum of Eq. (60), one obtains

$$\frac{{\partial \left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } \\ {z^{{\left( \varvec{k} \right)}} } \\ \end{array} } \right)_{il} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }} = \delta_{ip}^{{\left( \varvec{k} \right)}} \delta_{hl}^{{\left( \varvec{k} \right)}} \quad {\text{with}}\;\; \delta_{ip}^{{\left( \varvec{k} \right)}} = \left( {\begin{array}{*{20}c} {0^{{\left( \varvec{k} \right)}} } \\ {I^{{\left( \varvec{k} \right) \circ }} } \\ \end{array} } \right)_{ip} \;\;{\text{and}}\;\;\delta_{hl}^{{\left( \varvec{k} \right)}} = I^{{\left( \varvec{k} \right) \bullet }}_{hl}$$

(62)

By doing the same on the more complex derivate matrix terms in the second sum of Eq. (60), one obtains

$$\begin{aligned} & \frac{{\partial \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{ll^{{\prime }} }} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }} \\ & \quad = \mathop \sum \limits_{a,b = 1}^{{N_{\varvec{k}} ,N_{\varvec{k}} }} - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{la} \frac{{\partial \left[ {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right]_{ab} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }}\left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{bl^{{\prime }} }} \\ & \quad = \mathop \sum \limits_{a,b = 1}^{{N_{\varvec{k}} , N_{\varvec{k}} }} - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{la} \left[ {\frac{{\partial \left[ {I^{{\left( \varvec{k} \right) \bullet }} } \right]_{ab} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }} + \frac{{\partial \left[ {z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right]_{ab} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }}} \right] \times \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{bl^{{\prime }} }} \\ & \quad = \mathop \sum \limits_{a,b = 1}^{{N_{\varvec{k}} ,N_{\varvec{k}} }} - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{la} \left[ {\frac{{\partial \left( {\mathop \sum \nolimits_{{q = N_{\varvec{k}} + 1}}^{{K_{\varvec{k}} }} \left( {z^{{\left( \varvec{k} \right)\dag }} } \right)_{aq} z_{qb}^{{\left( \varvec{k} \right)}} } \right)}}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }}} \right]\left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{bl^{{\prime }} }} \\ & \quad = \mathop \sum \limits_{a,b = 1}^{{N_{\varvec{k}} ,N_{\varvec{k}} }} - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{la} \left\{ {\mathop \sum \limits_{q = 1}^{{K_{\varvec{k}} }} \left[ {\frac{{\partial \left( {z^{{\left( \varvec{k} \right)\dag }} } \right)_{aq} }}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }} \cdot z_{qb}^{{\left( \varvec{k} \right)}} + \left( {z^{{\left( \varvec{k} \right)\dag }} } \right)_{aq} \frac{{\partial \left( {z_{qb}^{{\left( \varvec{k} \right)}} } \right)}}{{\partial z_{ph}^{{\left( \varvec{k} \right)}} }}} \right]} \right\} \times \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{bl^{{\prime }} }} \\ & \quad = \mathop \sum \limits_{a,b = 1}^{{N_{\varvec{k}} ,N_{\varvec{k}} }} - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{la} \left\{ {\mathop \sum \limits_{q = 1}^{{K_{\varvec{k}} }} \left[ {\left( {z^{{\left( \varvec{k} \right)\dag }} } \right)_{aq} \delta_{qp}^{{\left( \varvec{k} \right)}} \delta_{hb}^{{\left( \varvec{k} \right)}} } \right]} \right\} \times \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{bl^{{\prime }} }} \\ & \quad = \mathop \sum \limits_{a,b = 1}^{{N_{\varvec{k}} ,N_{\varvec{k}} }} - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{la} \left[ {\left( {z^{{\left( \varvec{k} \right)\dag }} } \right)_{ap} \cdot \delta_{hb}^{{\left( \varvec{k} \right)}} } \right]\left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{bl^{{\prime }} }} \\ & \quad = - \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} I^{ \bullet } + z^{\dag } z} \right)^{ - 1} z^{{\left( \varvec{k} \right)\dag }} } \right]_{lp} \left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{{hl^{{\prime }} }} \\ \end{aligned}$$

(63)

By setting Eqs. (62) and (63) into Eq. (60), one obtains

$$\begin{aligned} & \frac{{\partial \varGamma_{ij}^{{\left( {kk} \right)}} \left( {z^{*} ,z} \right)}}{{\partial z_{\text{ph}}^{\left( k \right)} }} \\ & \quad = \mathop \sum \limits_{{l,l^{\prime } = 1}}^{{N_{k} ,N_{k} }} \left( {\begin{array}{*{20}c} {0^{\left( k \right) \circ } } \\ {I^{\left( k \right) \circ } } \\ \end{array} } \right)_{ip} I^{\left( k \right) \bullet }_{hl} \left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]_{{ll^{\prime } }} \left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)_{{l^{\prime } j}} \\ & \quad \quad - \,\mathop \sum \limits_{{l,l^{\prime } = 1}}^{{N_{k} ,N_{k} }} \left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } \\ {z^{\left( k \right)} } \\ \end{array} } \right)_{il} \left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \right]_{lp} \times \left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]_{{hl^{\prime } }} \left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)_{{l^{\prime } j}} \\ & \quad = \mathop \sum \limits_{l = 1}^{{N_{k} }} \left( {\begin{array}{*{20}c} {0^{\left( k \right) \circ } } \\ {I^{\left( k \right) \circ } } \\ \end{array} } \right)_{ip} \left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]_{hl} \left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)_{lj} \\ & \quad \quad - \left[ {\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } \\ {z^{\left( k \right)} } \\ \end{array} } \right)\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \right]} \right]_{ip} \left[ {\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right]_{hj} \\ & \quad = \left( {\begin{array}{*{20}c} {0^{\left( k \right) \circ } } \\ {I^{\left( k \right) \circ } } \\ \end{array} } \right)_{ip} \left[ {\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right]_{hj} \\ & \quad \quad - \,\left[ {\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } \\ {z^{\left( k \right)} } \\ \end{array} } \right)\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \right]} \right]_{ip} \left[ {\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right]_{hj} \\ & \quad = \left[ {\left( {\begin{array}{*{20}c} {0^{\left( k \right) \circ } } \\ {I^{\left( k \right) \circ } } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } \\ {z^{\left( k \right)} } \\ \end{array} } \right)\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \right]} \right]_{ip} \\ & \quad \quad \times \,\left[ {\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]\left( {\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right)} \right]_{hj} \\ & \quad = \left[ {\left( {\begin{array}{*{20}c} {0^{\left( k \right) \circ } } \\ {I^{\left( k \right) \circ } } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \\ {z^{\left( k \right)} \left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right]_{ip} \\ & \quad \quad \times \,\left[ {\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right]_{hj} \\ & \quad = \left( {\begin{array}{*{20}c} { - \left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \\ {I^{\left( k \right) \circ } - z^{\left( k \right)} \left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} z^{\left( k \right)\dag } } \\ \end{array} } \right)_{ip} \\ & \quad \quad \times \,\left[ {\left[ {\left( {I^{\left( k \right) \bullet } + z^{\left( k \right)\dag } z^{\left( k \right)} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{\left( k \right) \bullet } } & {z^{\left( k \right)\dag } } \\ \end{array} } \right)} \right]_{hj} \\ \end{aligned}$$

(64)

The matrix of the first element in the last line satisfies the identity (cf. Eq. 2.28 of Ref. [7])

$$\left( {\begin{array}{*{20}c} { - \,z^{{\left( \varvec{k} \right)\dag }} \left( {I^{{\left( \varvec{k} \right) \circ }} + z^{{\left( \varvec{k} \right)}} z^{{\left( \varvec{k} \right)\dag }} } \right)^{ - 1} } \\ {\left( {I^{{\left( \varvec{k} \right) \circ }} + z^{{\left( \varvec{k} \right)}} z^{{\left( \varvec{k} \right)\dag }} } \right)^{ - 1} } \\ \end{array} } \right) = \left[ {\left( {\begin{array}{*{20}c} { - z^{{\left( \varvec{k} \right)\dag }} } \\ {I^{{\left( \varvec{k} \right) \circ }} } \\ \end{array} } \right)\left( {I^{{\left( \varvec{k} \right) \circ }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]$$

(65)

By setting the above identity into Eq. (64), one finally obtains

$$\begin{aligned} & \frac{{\partial \varGamma_{ij}^{{\left( {\varvec{kk}} \right)}} \left( {\varvec{z}^{*} ,\varvec{z}} \right)}}{{\partial z_{\text{ph}}^{{\left( \varvec{k} \right)}} }} \\ & \quad = \left( {\begin{array}{*{20}c} { - \,z^{{\left( \varvec{k} \right)\dag }} \left( {I^{{\left( \varvec{k} \right) \circ }} + z^{{\left( \varvec{k} \right)}} z^{{\left( \varvec{k} \right)\dag }} } \right)^{ - 1} } \\ {\left( {I^{{\left( \varvec{k} \right) \circ }} + z^{{\left( \varvec{k} \right)}} z^{{\left( \varvec{k} \right)\dag }} } \right)^{ - 1} } \\ \end{array} } \right)_{ip} \left[ {\left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } & {z^{{\left( \varvec{k} \right)\dag }} } \\ \end{array} } \right)} \right]_{hj} \\ & \quad = \left[ {\left( {\begin{array}{*{20}c} { - z^{{\left( \varvec{k} \right)\dag }} } \\ {I^{{\left( \varvec{k} \right) \circ }} } \\ \end{array} } \right)\left( {I^{{\left( \varvec{k} \right) \circ }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]_{ip} \left[ {\left[ {\left( {I^{{\left( \varvec{k} \right) \bullet }} + z^{{\left( \varvec{k} \right)\dag }} z^{{\left( \varvec{k} \right)}} } \right)^{ - 1} } \right]\left( {\begin{array}{*{20}c} {I^{{\left( \varvec{k} \right) \bullet }} } & {z^{{\left( \varvec{k} \right)\dag }} } \\ \end{array} } \right)} \right]_{hj} \\ \end{aligned}$$

(66)