import torch
from ...utils import fast_power
[docs]
def radial_slater(
R, bas_n, bas_exp, xyz=None, derivative=0, sum_grad=True, sum_hess=True
):
"""Compute the radial part of STOs (or its derivative).
.. math:
sto = r^n exp(-\alpha |r|)
Args:
R (torch.tensor): distance between each electron and each atom
bas_n (torch.tensor): principal quantum number
bas_exp (torch.tensor): exponents of the exponential
Keyword Arguments:
xyz (torch.tensor): positions of the electrons
(needed for derivative) (default: {None})
derivative (int): degree of the derivative (default: {0})
0 : value of the function
1 : first derivative
2 : pure second derivative
3 : mixed second derivative
sum_grad (bool): return the sum_grad, i.e the sum of the gradients
(default: {True})
sum_hess (bool): return the sum_hess, i.e the sum of the diag hessian
(default: {False})
mixed_hess (bool): return the full hessian for each electron
i.e. dxdy dxdz dydz ... mixed derivatives
(default: {False})
Returns:
torch.tensor: values of each orbital radial part at each position
"""
if not isinstance(derivative, list):
derivative = [derivative]
def _kernel():
"""Return the kernel."""
return rn * er
def _first_derivative_kernel():
"""Return the first derivative."""
if sum_grad:
nabla_rn_sum = nabla_rn.sum(3)
nabla_er_sum = nabla_er.sum(3)
return nabla_rn_sum * er + rn * nabla_er_sum
else:
return nabla_rn * er.unsqueeze(-1) + rn.unsqueeze(-1) * nabla_er
def _second_derivative_kernel():
"""Return the pure second derivative i.e. d^2/dx^2"""
if sum_hess:
lap_rn = nRnm2 * (bas_n + 1)
lap_er = bexp_er * (bas_exp - 2.0 / R)
return lap_rn * er + 2 * (nabla_rn * nabla_er).sum(3) + rn * lap_er
else:
xyz2 = xyz * xyz
xyz2 = xyz2 / xyz2.sum(-1, keepdim=True)
lap_rn = nRnm2.unsqueeze(-1) * (1.0 + (bas_n - 2).unsqueeze(-1) * xyz2)
lap_er = bexp_er.unsqueeze(-1) * (
bas_exp.unsqueeze(-1) * xyz2 + (-1 + xyz2) / R.unsqueeze(-1)
)
return (
lap_rn * er.unsqueeze(-1)
+ 2 * (nabla_rn * nabla_er)
+ rn.unsqueeze(-1) * lap_er
)
def _mixed_second_derivative_kernel():
"""Returns the mixed second derivative i.e. d^2/dxdy.
where x and y are coordinate of the same electron."""
mix_prod = xyz[..., [[0, 1], [0, 2], [1, 2]]].prod(-1)
nRnm4 = nRnm2 / (xyz * xyz).sum(-1)
lap_rn = ((bas_n - 2) * nRnm4).unsqueeze(-1) * mix_prod
lap_er = (
(bexp_er / (xyz * xyz).sum(-1)).unsqueeze(-1)
* mix_prod
* (bas_exp.unsqueeze(-1) + 1.0 / R.unsqueeze(-1))
)
return (
lap_rn * er.unsqueeze(-1)
+ (
nabla_rn[..., [[0, 1], [0, 2], [1, 2]]]
* nabla_er[..., [[1, 0], [2, 0], [2, 1]]]
).sum(-1)
+ rn.unsqueeze(-1) * lap_er
)
# computes the basic quantities
rn = fast_power(R, bas_n)
er = torch.exp(-bas_exp * R)
# computes the grad
if any(x in derivative for x in [1, 2, 3]):
Rnm2 = R ** (bas_n - 2)
nRnm2 = bas_n * Rnm2
bexp_er = bas_exp * er
nabla_rn = (nRnm2).unsqueeze(-1) * xyz
nabla_er = -(bexp_er).unsqueeze(-1) * xyz / R.unsqueeze(-1)
return return_required_data(
derivative,
_kernel,
_first_derivative_kernel,
_second_derivative_kernel,
_mixed_second_derivative_kernel,
)
[docs]
def radial_gaussian(
R, bas_n, bas_exp, xyz=None, derivative=[0], sum_grad=True, sum_hess=True
):
"""Compute the radial part of GTOs (or its derivative).
.. math:
gto = r ^ n exp(-\alpha r ^ 2)
Args:
R(torch.tensor): distance between each electron and each atom
bas_n(torch.tensor): principal quantum number
bas_exp(torch.tensor): exponents of the exponential
Keyword Arguments:
xyz(torch.tensor): positions of the electrons
(needed for derivative)(default: {None})
derivative(int): degree of the derivative(default: {0})
sum_grad(bool): return the sum_grad, i.e the sum of the gradients
(default: {True})
Returns:
torch.tensor: values of each orbital radial part at each position
"""
if not isinstance(derivative, list):
derivative = [derivative]
def _kernel():
return rn * er
def _first_derivative_kernel():
if sum_grad:
nabla_rn_sum = nabla_rn.sum(3)
nabla_er_sum = nabla_er.sum(3)
return nabla_rn_sum * er + rn * nabla_er_sum
else:
return nabla_rn * er.unsqueeze(-1) + rn.unsqueeze(-1) * nabla_er
def _second_derivative_kernel():
if sum_hess:
lap_rn = nRnm2 * (bas_n + 1)
lap_er = bas_exp * er * (4 * bas_exp * R2 - 6)
return lap_rn * er + 2 * (nabla_rn * nabla_er).sum(3) + rn * lap_er
else:
xyz2 = xyz * xyz
lap_er = (bas_exp * er).unsqueeze(-1) * (
4 * bas_exp.unsqueeze(-1) * xyz2 - 2
)
xyz2 = xyz2 / xyz2.sum(-1, keepdim=True)
lap_rn = nRnm2.unsqueeze(-1) * (1.0 + (bas_n - 2).unsqueeze(-1) * xyz2)
return (
lap_rn * er.unsqueeze(-1)
+ 2 * (nabla_rn * nabla_er)
+ rn.unsqueeze(-1) * lap_er
)
def _mixed_second_derivative_kernel():
"""Returns the mixed second derivative i.e. d^2/dxdy.
where x and y are coordinate of the same electron."""
mix_prod = xyz[..., [[0, 1], [0, 2], [1, 2]]].prod(-1)
nRnm4 = nRnm2 / (xyz * xyz).sum(-1)
lap_rn = ((bas_n - 2) * nRnm4).unsqueeze(-1) * mix_prod
lap_er = 4 * (bexp_er * bas_exp).unsqueeze(-1) * mix_prod
return (
lap_rn * er.unsqueeze(-1)
+ (
nabla_rn[..., [[0, 1], [0, 2], [1, 2]]]
* nabla_er[..., [[1, 0], [2, 0], [2, 1]]]
).sum(-1)
+ rn.unsqueeze(-1) * lap_er
)
# computes the basic quantities
R2 = R * R
rn = fast_power(R, bas_n)
er = torch.exp(-bas_exp * R2)
# computes the grads
if any(x in derivative for x in [1, 2, 3]):
Rnm2 = R ** (bas_n - 2)
nRnm2 = bas_n * Rnm2
bexp_er = bas_exp * er
nabla_rn = (nRnm2).unsqueeze(-1) * xyz
nabla_er = -2 * (bexp_er).unsqueeze(-1) * xyz
return return_required_data(
derivative,
_kernel,
_first_derivative_kernel,
_second_derivative_kernel,
_mixed_second_derivative_kernel,
)
[docs]
def radial_gaussian_pure(
R, bas_n, bas_exp, xyz=None, derivative=[0], sum_grad=True, sum_hess=True
):
"""Compute the radial part of GTOs (or its derivative).
.. math:
gto = exp(-\alpha r ^ 2)
Args:
R(torch.tensor): distance between each electron and each atom
bas_n(torch.tensor): principal quantum number
bas_exp(torch.tensor): exponents of the exponential
Keyword Arguments:
xyz(torch.tensor): positions of the electrons
(needed for derivative)(default: {None})
derivative(int): degree of the derivative(default: {0})
sum_grad(bool): return the sum_grad, i.e the sum of the gradients
(default: {True})
sum_hess(bool): return the sum_hess, i.e the sum of the lapacian
(default: {True})
Returns:
torch.tensor: values of each orbital radial part at each position
"""
if not isinstance(derivative, list):
derivative = [derivative]
def _kernel():
return er
def _first_derivative_kernel():
if sum_grad:
return nabla_er.sum(3)
else:
return nabla_er
def _second_derivative_kernel():
if sum_hess:
lap_er = bas_exp * er * (4 * bas_exp * R2 - 6)
return lap_er
else:
xyz2 = xyz * xyz
lap_er = (bas_exp * er).unsqueeze(-1) * (
4 * bas_exp.unsqueeze(-1) * xyz2 - 2
)
return lap_er
def _mixed_second_derivative_kernel():
"""Returns the mixed second derivative i.e. d^2/dxdy.
where x and y are coordinate of the same electron."""
mix_prod = xyz[..., [[0, 1], [0, 2], [1, 2]]].prod(-1)
lap_er = 4 * (bexp_er * bas_exp).unsqueeze(-1) * mix_prod
return lap_er
# computes the basic quantities
R2 = R * R
er = torch.exp(-bas_exp * R2)
# computes the grads
if any(x in derivative for x in [1, 2, 3]):
bexp_er = bas_exp * er
nabla_er = -2 * (bexp_er).unsqueeze(-1) * xyz
return return_required_data(
derivative,
_kernel,
_first_derivative_kernel,
_second_derivative_kernel,
_mixed_second_derivative_kernel,
)
[docs]
def radial_slater_pure(
R, bas_n, bas_exp, xyz=None, derivative=0, sum_grad=True, sum_hess=True
):
"""Compute the radial part of STOs (or its derivative).
.. math:
sto = exp(-\alpha | r |)
Args:
R(torch.tensor): distance between each electron and each atom
bas_n(torch.tensor): principal quantum number
bas_exp(torch.tensor): exponents of the exponential
Keyword Arguments:
xyz(torch.tensor): positions of the electrons
(needed for derivative)(default: {None})
derivative(int): degree of the derivative(default: {0})
sum_grad(bool): return the sum_grad, i.e the sum of the gradients
(default: {True})
sum_hess(bool): return the sum_hess, i.e the sum of the laplacian
(default: {True})
Returns:
torch.tensor: values of each orbital radial part at each position
"""
if not isinstance(derivative, list):
derivative = [derivative]
def _kernel():
return er
def _first_derivative_kernel():
if sum_grad:
return nabla_er.sum(3)
else:
return nabla_er
def _second_derivative_kernel():
if sum_hess:
return bexp_er * (bas_exp - 2.0 / R)
else:
xyz2 = xyz * xyz / (R * R).unsqueeze(-1)
lap_er = bexp_er.unsqueeze(-1) * (
bas_exp.unsqueeze(-1) * xyz2 - (1 - xyz2) / R.unsqueeze(-1)
)
return lap_er
def _mixed_second_derivative_kernel():
"""Returns the mixed second derivative i.e. d^2/dxdy.
where x and y are coordinate of the same electron."""
mix_prod = xyz[..., [[0, 1], [0, 2], [1, 2]]].prod(-1)
lap_er = (
(bexp_er / (xyz * xyz).sum(-1)).unsqueeze(-1)
* mix_prod
* (bas_exp.unsqueeze(-1) + 1.0 / R.unsqueeze(-1))
)
return lap_er
# computes the basic gradients
er = torch.exp(-bas_exp * R)
# computes the grad
if any(x in derivative for x in [1, 2, 3]):
bexp_er = bas_exp * er
nabla_er = -(bexp_er).unsqueeze(-1) * xyz / R.unsqueeze(-1)
return return_required_data(
derivative,
_kernel,
_first_derivative_kernel,
_second_derivative_kernel,
_mixed_second_derivative_kernel,
)
[docs]
def return_required_data(
derivative,
_kernel,
_first_derivative_kernel,
_second_derivative_kernel,
_mixed_second_derivative_kernel,
):
"""Returns the data contained in derivative
Args:
derivative(list): list of the derivatives required
_kernel(callable): kernel of the values
_first_derivative_kernel(callable): kernel for 1st der
_second_derivative_kernel(callable): kernel for 2nd der
Returns:
list: values of the different der requried
"""
# prepare the output/kernel
output = []
fns = [
_kernel,
_first_derivative_kernel,
_second_derivative_kernel,
_mixed_second_derivative_kernel,
]
# compute the requested functions
for d in derivative:
output.append(fns[d]())
if len(derivative) == 1:
return output[0]
else:
return output