Wavefunction
Wavefunction in the Slater-Jastrow-backflow (SJB) form is represented by the casino.Wfn class.
\[\Psi(\mathbf{r}) = e^{J(\mathbf{r})}\Phi(\mathbf{X}(\mathbf{r}))\]
where \(\mathbf{X}(\mathbf{r}) = \mathbf{r} + \xi(\mathbf{r})\) are the collective coordinates, which depend on the configuration of the whole system.
Summary of Methods
Wfn class has the following methods:
value
Value of wavefunction:
\[\Psi(\mathbf{r}) = e^{J(\mathbf{r})}\Phi(\mathbf{r} + \xi(\mathbf{r}))\]
coulomb
Value of e-e, e-n and n-n coulomb interaction:
\[V_{coul} = \sum_i^{N_e} \sum_{j \gt i}^{N_e} \frac{1}{r_{ij}} - \sum_i^{N_e} \sum_\alpha^{N_I} \frac{Z_\alpha}{r_{i\alpha}} + \sum_\alpha^{N_I} \sum_{\beta \gt \alpha}^{N_I} \frac{Z_\alpha Z_\beta}{r_{\alpha\beta}}\]
nonlocal_potential
Value of nonlocal potential:
\[V_{nl} = \sum_i^{N_e} \sum_l \frac{2l + 1}{4\pi} V_l(r_i) \int_{4\pi} P_l(cos \ \theta'_{i}) \frac{\Psi(\mathbf{r}\rvert_{r_i \to r_i'})}{\Psi(\mathbf{r})} d\Omega'_i\]
where \(P_l\) denotes a Legendre polynomial, \(V_l(r_i)\) is pseudopotential and \(\theta'_{i}=\angle(r_i, r_i')\).
Integral of the function \(f\) defined on the unit sphere is approximated as:
\[\frac{1}{4\pi} \int_{4\pi} f(\Omega) d\Omega \approx \sum_q c_q f(r_q)\]
The values of the coefficients \(c_q\) and unit vectors \(r_q\) can be found in the appendix to the article
Nonlocal pseudopotentials and diffusion Monte Carlo
after which we obtain:
\[V_{nl} = \sum_i^{N_e} \sum_l (2l + 1) V_l(r_i) \sum_q c_q P_l(cos \ \theta'_{i,q}) \frac{\Psi(\mathbf{r}\rvert_{r_i \to r_{i,q}'})}{\Psi(\mathbf{r})}\]
kinetic_energy
Value of kinetic energy usually represented as the sum of two terms:
\[T_{kin} = -\frac{1}{2} \sum_i^{N_e} \frac{\nabla_i^2 \Psi}{\Psi} = \sum_i^{N_e} \left( 2T_i - F_i^2 \right)\]
where \(T_i\) and \(F_i\) are expressed through the logarithm of the modulus of the wave function:
\[T_i = - \frac{1}{4} \nabla_i^2 \ln \Psi = - \frac{1}{4} \left[ \frac{\nabla_i^2 \Psi}{\Psi} - \left(\frac{\nabla_i \Psi}{\Psi}\right)^2 \right]\]
\[F_i = \frac{1}{\sqrt{2}} \nabla_i \ln \Psi\]
This is convenient because the wave function is a product which leads to the following expressions:
\[T_i = - \frac{1}{4} \left[ \frac{\nabla_i^2 \Phi}{\Phi} - \left(\frac{\nabla_i \Phi}{\Phi}\right)^2 + \nabla_i^2 J \right]\]
\[F_i = \frac{1}{\sqrt{2}} \left( \frac{\nabla_i \Phi}{\Phi} + \nabla_i J \right)\]
if backflow displacement \(\xi(\mathbf{r})\) is not zero the coordinate transformation must be taken into account:
\[\nabla \Phi = \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\]
\[\nabla^2 \Phi = tr\left(\left(\frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)^T
\cdot \frac{\partial^2 \Phi}{\partial^2 \mathbf{X}}
\cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)
+ \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial^2 \mathbf{X}}{\partial^2 \mathbf{r}}\]
energy
Value of local energy.
\[E_L = T_{kin} + V_{coul} + V_{nl}\]
drift_velocity
Drift velocity (one half of the quantum force):
\[\mathbf{v} = \nabla \ln \Psi = \nabla J + \frac{\nabla \Phi}{\Phi}\]
if backflow displacement \(\xi(\mathbf{r})\) is not zero the coordinate transformation must be taken into account:
\[\nabla \Phi = \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\]
value_parameters_d1
Partial derivatives of the wave function with respect to the Jastrow, backflow and Slater parameters \(\{\alpha_i^J, \alpha_i^B, \alpha_i^S \}\):
\[\frac{\partial \ln \Psi}{\partial \alpha} = \left(\frac{\partial J}{\partial \alpha_i^J} , \frac{\nabla_{\mathbf{X}} \Phi}{\Phi} \cdot \frac{\partial \mathbf{X}}{\partial \alpha_i^B} , \frac{\partial \ln \Phi}{\partial \alpha_i^S} \right)\]
kinetic_energy_parameters_d1
Partial derivatives of the kinetic energy with respect to the Jastrow, backflow and Slater parameters \(\{\alpha_i^J, \alpha_i^B, \alpha_i^S \}\):
\[\frac{\partial T_{kin}}{\partial \alpha} = -\frac{1}{2} \sum_i^{N_e} \frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha} = -\frac{1}{2} \sum_i^{N_e} \left(
\frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha^J},
\frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha^B},
\frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha^S},
\right)\]
\[\frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha^J} = \frac{\partial \nabla^2 J}{\partial \alpha_i^J} + 2 \frac{\nabla \Psi}{\Psi} \cdot \frac{\partial \nabla J}{\partial \alpha_i^J},\]
\[\frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha^B} =
\frac{\partial}{\partial \alpha_i^B} \left( \frac{\nabla^2 \Phi}{\Phi} \right) +
2 \nabla J \cdot \frac{\partial}{\partial \alpha_i^B} \left( \frac{\nabla \Phi}{\Phi} \right)\]
since backflow parameters enter \(\Psi\) only through the collective coordinates \(\mathbf{X}(\mathbf{r})\),
these derivatives are obtained by the chain rule:
\[\frac{\partial \nabla \Phi}{\partial \alpha_i^B} =
\frac{\partial \mathbf{X}}{\partial \alpha_i^B} \cdot \frac{\partial^2 \Phi}{\partial^2 \mathbf{X}}
\cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}
+ \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial^2 \mathbf{X}}{\partial \alpha_i^B \partial \mathbf{r}}\]
\[\frac{\partial \nabla^2 \Phi}{\partial \alpha_i^B} =
tr\left(\left(\frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)^T
\cdot \left(\frac{\partial \mathbf{X}}{\partial \alpha_i^B} \cdot \frac{\partial^3 \Phi}{\partial^3 \mathbf{X}}\right)
\cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)
+ 2 \, tr\left(\left(\frac{\partial^2 \mathbf{X}}{\partial \alpha_i^B \partial \mathbf{r}}\right)^T
\cdot \frac{\partial^2 \Phi}{\partial^2 \mathbf{X}}
\cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)
+ \frac{\partial \mathbf{X}}{\partial \alpha_i^B} \cdot \frac{\partial^2 \Phi}{\partial^2 \mathbf{X}}
\cdot \frac{\partial^2 \mathbf{X}}{\partial^2 \mathbf{r}}
+ \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial^3 \mathbf{X}}{\partial \alpha_i^B \partial^2 \mathbf{r}}\]
note that the third derivatives of the Slater part \(\partial^3 \Phi / \partial^3 \mathbf{X}\) are required,
which makes backflow optimization the most expensive operation.
\[\frac{\partial \nabla_i^2 \Psi / \Psi}{\partial \alpha^S} = \frac{\partial \nabla^2 \ln \Phi}{\partial \alpha^S} +
2 \frac{\partial \nabla \ln \Phi}{\partial \alpha^S} \left[ \nabla J + \frac{\nabla \Phi}{\Phi} \right]\]
if backflow displacement \(\xi(\mathbf{r})\) is not zero the coordinate transformation must be taken into account:
\[\nabla \Phi = \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\]
\[\nabla^2 \Phi = tr\left(\left(\frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)^T
\cdot \frac{\partial^2 \Phi}{\partial^2 \mathbf{X}}
\cdot \frac{\partial \mathbf{X}}{\partial \mathbf{r}}\right)
+ \frac{\partial \Phi}{\partial \mathbf{X}} \cdot \frac{\partial^2 \mathbf{X}}{\partial^2 \mathbf{r}}\]
nonlocal_energy_parameters_d1
Partial derivatives of the nonlocal energy with respect to the Jastrow, backflow and Slater parameters:
\[\frac{\partial V_{nl}}{\partial \alpha} = \sum_i^{N_e} \sum_l (2l + 1) V_l(r_i) \sum_q c_q P_l \left( cos \ \theta'_{i,q} \right) \frac{\partial}{\partial \alpha}
\left[ \frac{\Psi(\mathbf{r}\rvert_{r_i \to r_{i,q}'})}{\Psi(\mathbf{r})} \right]\]
\[= \sum_i^{N_e} \sum_l (2l + 1) V_l(r_i) \sum_q c_q P_l \left( cos \ \theta'_{i,q} \right) \frac{\Psi(\mathbf{r}\rvert_{r_i \to r_i'})}{\Psi(\mathbf{r})}
\left[ \frac{\partial \ln \Psi(\mathbf{r}\rvert_{r_i \to r_{i,q}'})}{\partial \alpha} - \frac{\partial \ln \Psi(\mathbf{r})}{\partial \alpha} \right]\]
energy_parameters_d1
Partial derivatives of the local energy with respect to the parameters.
The Coulomb term does not depend on the parameters, so:
\[\frac{\partial E_L}{\partial \alpha} = \frac{\partial T_{kin}}{\partial \alpha} + \frac{\partial V_{nl}}{\partial \alpha}\]