Alternating Linear approximation of the ISA (aLISA) method
Non-linear optimization problem
[1]:
import numpy as np
from setup import prepare_argument_dict, prepare_grid_and_dens, print_results
from horton_part import ExpBasisFuncHelper, LinearISAWPart
Convex optimization method
[2]:
mol, grid, rho = prepare_grid_and_dens("data/h2o.fchk")
def use_cvxopt_solver():
"""Local LISA by solving convex optimization problem."""
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = "cvxopt"
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
use_cvxopt_solver()
The number of electrons: 10.000003764139395
Coordinates of the atoms
[[ 0. 0. 0.224]
[-0. 1.457 -0.896]
[-0. -1.457 -0.896]]
Atomic numbers of the atom
[8 1 1]
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : CVXOPT
Basis function type : GAUSS
Grid type : 1
Load GAUSS basis functions
Iteration Change Entropy
1 1.22856e-01 2.11396e-01
2 2.65514e-02 1.08102e-01
3 1.49501e-02 7.80048e-02
4 9.43136e-03 6.28961e-02
5 6.28278e-03 5.43378e-02
6 4.36240e-03 4.91980e-02
7 3.13687e-03 4.59887e-02
8 2.32436e-03 4.39260e-02
9 1.76689e-03 4.25692e-02
10 1.37223e-03 4.16592e-02
11 1.08483e-03 4.10387e-02
12 8.70229e-04 4.06094e-02
13 7.06492e-04 4.03086e-02
14 5.79250e-04 4.00953e-02
15 4.78831e-04 3.99426e-02
16 3.98556e-04 3.98323e-02
17 3.33687e-04 3.97519e-02
18 2.80794e-04 3.96929e-02
19 2.37332e-04 3.96492e-02
20 2.01386e-04 3.96168e-02
21 1.71486e-04 3.95926e-02
22 1.46492e-04 3.95743e-02
23 1.25506e-04 3.95606e-02
24 1.07816e-04 3.95501e-02
25 9.28501e-05 3.95422e-02
26 8.01470e-05 3.95361e-02
27 6.93317e-05 3.95315e-02
28 6.00975e-05 3.95279e-02
29 5.21920e-05 3.95251e-02
30 4.54072e-05 3.95230e-02
31 3.95702e-05 3.95214e-02
32 3.45374e-05 3.95201e-02
33 3.01887e-05 3.95191e-02
34 2.64235e-05 3.95183e-02
35 2.31572e-05 3.95177e-02
36 2.03187e-05 3.95173e-02
37 1.78476e-05 3.95169e-02
38 1.56930e-05 3.95166e-02
39 1.38115e-05 3.95164e-02
40 1.21661e-05 3.95162e-02
41 1.07252e-05 3.95161e-02
42 9.46198e-06 3.95160e-02
43 8.35316e-06 3.95159e-02
44 7.37884e-06 3.95158e-02
45 6.52185e-06 3.95158e-02
46 5.76738e-06 3.95158e-02
47 5.10260e-06 3.95157e-02
48 4.51639e-06 3.95157e-02
49 3.99910e-06 3.95157e-02
50 3.54233e-06 3.95157e-02
51 3.13875e-06 3.95157e-02
52 2.78197e-06 3.95157e-02
53 2.46641e-06 3.95156e-02
54 2.18717e-06 3.95156e-02
55 1.93998e-06 3.95156e-02
56 1.72107e-06 3.95156e-02
57 1.52713e-06 3.95156e-02
58 1.35528e-06 3.95156e-02
59 1.20294e-06 3.95156e-02
60 1.06787e-06 3.95156e-02
61 9.48077e-07 3.95156e-02
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.881 0.441 0.441]
cartesian multipoles:
[[-0.881 0. -0. 0.133 -5.381 -0. -0. -4.802 -0. -5.094 0. 0. 0.034 0. -0. 0. -0. 0.416 -0. 0.367]
[ 0.441 0. 0.022 -0.001 -0.275 -0. -0. -0.257 0.002 -0.26 0. 0.008 0.005 0. 0. 0. 0.024 0.009 0.009 0.023]
[ 0.441 0. -0.022 -0.001 -0.275 0. 0. -0.257 -0.002 -0.26 0. -0.008 0.005 0. -0. 0. -0.024 0.009 -0.009 0.023]]
Trust-region method
[3]:
def use_trust_region_implicit_constr():
"""Local LISA by solving convex optimization problem."""
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = "trust-region"
kwargs["solver_options"] = {"explicit_constr": False}
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
use_trust_region_implicit_constr()
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : TRUST-REGION
Basis function type : GAUSS
Grid type : 1
Solver options -- explicit_constr : False
Load GAUSS basis functions
Iteration Change Entropy
1 1.22857e-01 2.11396e-01
/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py:317: UserWarning: delta_grad == 0.0. Check if the approximated function is linear. If the function is linear better results can be obtained by defining the Hessian as zero instead of using quasi-Newton approximations.
self.H.update(self.x - self.x_prev, self.g - self.g_prev)
2 2.65512e-02 1.08101e-01
3 1.49500e-02 7.80036e-02
4 9.43166e-03 6.28912e-02
5 6.28281e-03 5.43367e-02
6 4.36175e-03 4.91978e-02
7 3.13726e-03 4.59789e-02
8 2.32440e-03 4.39258e-02
9 1.76692e-03 4.25672e-02
10 1.37216e-03 4.16575e-02
11 1.08463e-03 4.10377e-02
12 8.69893e-04 4.06033e-02
13 7.06705e-04 4.02989e-02
14 5.79286e-04 4.00944e-02
15 4.78943e-04 3.99409e-02
16 3.98627e-04 3.98319e-02
17 3.33627e-04 3.97500e-02
18 2.80759e-04 3.96915e-02
19 2.37306e-04 3.96441e-02
20 2.01458e-04 3.96154e-02
21 1.71255e-04 3.95904e-02
22 1.46888e-04 3.95721e-02
23 1.25132e-04 3.95594e-02
24 1.07558e-04 3.95400e-02
25 9.26409e-05 3.95368e-02
26 8.02243e-05 3.95259e-02
27 6.92730e-05 3.95295e-02
28 6.00420e-05 3.95180e-02
29 5.22332e-05 3.95232e-02
30 4.53907e-05 3.95170e-02
31 3.96587e-05 3.95199e-02
32 3.45903e-05 3.95146e-02
33 3.03753e-05 3.95168e-02
34 2.66168e-05 3.95163e-02
35 2.26949e-05 3.95078e-02
36 1.98542e-05 3.95071e-02
37 1.85424e-05 3.95066e-02
38 1.57605e-05 3.95146e-02
39 1.44842e-05 3.95159e-02
40 1.23179e-05 3.95102e-02
41 1.09648e-05 3.95139e-02
42 9.72528e-06 3.95139e-02
43 8.27633e-06 3.95139e-02
44 7.38677e-06 3.95137e-02
45 8.24796e-06 3.95136e-02
46 5.54593e-06 3.95057e-02
47 5.16234e-06 3.95056e-02
48 5.00161e-06 3.95054e-02
49 3.69788e-06 3.95056e-02
50 3.47616e-06 3.95056e-02
51 3.25830e-06 3.95053e-02
52 2.46714e-06 3.95055e-02
53 2.48737e-06 3.95056e-02
54 1.06701e-05 3.95055e-02
55 1.14762e-05 3.95041e-02
56 1.46453e-06 3.95056e-02
57 1.41633e-06 3.95056e-02
58 1.34265e-06 3.95055e-02
59 1.06729e-06 3.95055e-02
60 1.92067e-06 3.95055e-02
61 2.34427e-06 3.95053e-02
62 2.26184e-06 3.95056e-02
63 6.45368e-07 3.95053e-02
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.881 0.441 0.441]
cartesian multipoles:
[[-0.881 0. 0. 0.133 -5.381 -0. -0. -4.801 -0. -5.094 -0. 0. 0.034 0. 0. 0. 0. 0.416 0. 0.367]
[ 0.441 0. 0.022 -0.001 -0.275 -0. -0. -0.257 0.002 -0.26 0. 0.008 0.005 0. 0. 0. 0.024 0.009 0.009 0.023]
[ 0.441 0. -0.022 -0.001 -0.275 0. 0. -0.257 -0.002 -0.26 0. -0.008 0.005 0. -0. 0. -0.024 0.009 -0.009 0.023]]
Non-linear equations (fixed-point equations)
Self-consistent method
[4]:
def use_sc_solver():
"""Self-consistent solver."""
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = "sc"
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
use_sc_solver()
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : SC
Basis function type : GAUSS
Grid type : 1
Load GAUSS basis functions
Iteration Change Entropy
1 1.22855e-01 2.11396e-01
2 2.65514e-02 1.08102e-01
3 1.49501e-02 7.80048e-02
4 9.43137e-03 6.28961e-02
5 6.28279e-03 5.43378e-02
6 4.36239e-03 4.91980e-02
7 3.13686e-03 4.59887e-02
8 2.32436e-03 4.39260e-02
9 1.76689e-03 4.25692e-02
10 1.37224e-03 4.16592e-02
11 1.08483e-03 4.10387e-02
12 8.70230e-04 4.06094e-02
13 7.06494e-04 4.03086e-02
14 5.79250e-04 4.00953e-02
15 4.78833e-04 3.99426e-02
16 3.98555e-04 3.98323e-02
17 3.33686e-04 3.97519e-02
18 2.80794e-04 3.96929e-02
19 2.37332e-04 3.96492e-02
20 2.01386e-04 3.96168e-02
21 1.71485e-04 3.95926e-02
22 1.46492e-04 3.95743e-02
23 1.25506e-04 3.95606e-02
24 1.07816e-04 3.95501e-02
25 9.28499e-05 3.95422e-02
26 8.01454e-05 3.95361e-02
27 6.93308e-05 3.95315e-02
28 6.00965e-05 3.95279e-02
29 5.21911e-05 3.95251e-02
30 4.54063e-05 3.95230e-02
31 3.95693e-05 3.95214e-02
32 3.45365e-05 3.95201e-02
33 3.01888e-05 3.95191e-02
34 2.64229e-05 3.95183e-02
35 2.31565e-05 3.95177e-02
36 2.03179e-05 3.95173e-02
37 1.78467e-05 3.95169e-02
38 1.56920e-05 3.95166e-02
39 1.38104e-05 3.95164e-02
40 1.21649e-05 3.95162e-02
41 1.07240e-05 3.95161e-02
42 9.46161e-06 3.95160e-02
43 8.35226e-06 3.95159e-02
44 7.37789e-06 3.95158e-02
45 6.52082e-06 3.95158e-02
46 5.76628e-06 3.95158e-02
47 5.10146e-06 3.95157e-02
48 4.51860e-06 3.95157e-02
49 3.99824e-06 3.95157e-02
50 3.54140e-06 3.95157e-02
51 3.13774e-06 3.95157e-02
52 2.78160e-06 3.95157e-02
53 2.46552e-06 3.95156e-02
54 2.18621e-06 3.95156e-02
55 1.93964e-06 3.95156e-02
56 1.72015e-06 3.95156e-02
57 1.52399e-06 3.95156e-02
58 1.35421e-06 3.95156e-02
59 1.20259e-06 3.95156e-02
60 1.06770e-06 3.95156e-02
61 9.46257e-07 3.95156e-02
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.881 0.441 0.441]
cartesian multipoles:
[[-0.881 0. -0. 0.133 -5.381 -0. -0. -4.802 -0. -5.094 -0. -0. 0.034 0. 0. 0. -0. 0.416 -0. 0.367]
[ 0.441 0. 0.022 -0.001 -0.275 -0. -0. -0.257 0.002 -0.26 0. 0.008 0.005 0. 0. 0. 0.024 0.009 0.009 0.023]
[ 0.441 0. -0.022 -0.001 -0.275 0. 0. -0.257 -0.002 -0.26 0. -0.008 0.005 0. -0. 0. -0.024 0.009 -0.009 0.023]]
Direct Inversion in Iterative Space (DIIS)
This method has been extensively used to solve self-consistent field (SCF) problems in the fields of quantum chemistry and physics. In this tutorial, we employ this method to accelerate the solving of fixed-point problems.
It should be noted that one potential issue with this method is that non-negative parameters cannot be guaranteed during optimization in the conventional DIIS approach. Although this issue can be addressed by explicitly introducing constraints to the linear combination coefficients, key concepts in DIIS, numerical issues may still arise, such as singular matrices or convergence problems.
[5]:
def use_diis_solver():
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = "diis"
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
use_diis_solver()
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : DIIS
Basis function type : GAUSS
Grid type : 1
Load GAUSS basis functions
Iteration Change Entropy
1 1.22855e-01 2.11396e-01
2 2.65514e-02 1.08102e-01
3 1.49501e-02 7.80048e-02
4 9.43139e-03 6.28961e-02
5 6.28279e-03 5.43378e-02
6 4.36240e-03 4.91980e-02
7 3.13687e-03 4.59888e-02
8 2.32435e-03 4.39260e-02
9 1.76690e-03 4.25692e-02
10 1.37224e-03 4.16592e-02
11 1.08483e-03 4.10387e-02
12 8.70225e-04 4.06094e-02
13 7.06490e-04 4.03086e-02
14 5.79247e-04 4.00953e-02
15 4.78829e-04 3.99426e-02
16 3.98555e-04 3.98323e-02
17 3.33687e-04 3.97519e-02
18 2.80792e-04 3.96929e-02
19 2.37332e-04 3.96492e-02
20 2.01386e-04 3.96168e-02
21 1.71486e-04 3.95926e-02
22 1.46496e-04 3.95743e-02
23 1.25504e-04 3.95606e-02
24 1.07814e-04 3.95501e-02
25 9.28271e-05 3.95422e-02
26 8.01196e-05 3.95361e-02
27 6.93741e-05 3.95315e-02
28 6.01082e-05 3.95279e-02
29 5.21961e-05 3.95251e-02
30 4.54061e-05 3.95230e-02
31 3.95715e-05 3.95214e-02
32 3.45348e-05 3.95201e-02
33 3.01724e-05 3.95191e-02
34 2.64375e-05 3.95183e-02
35 2.31613e-05 3.95177e-02
36 2.03145e-05 3.95173e-02
37 1.78539e-05 3.95169e-02
38 1.56953e-05 3.95166e-02
39 1.38110e-05 3.95164e-02
40 1.21668e-05 3.95162e-02
41 1.07249e-05 3.95161e-02
42 9.46136e-06 3.95160e-02
43 8.35235e-06 3.95159e-02
44 7.37938e-06 3.95158e-02
45 6.51932e-06 3.95158e-02
46 5.76641e-06 3.95158e-02
47 5.10103e-06 3.95157e-02
48 4.51502e-06 3.95157e-02
49 3.99851e-06 3.95157e-02
50 3.54139e-06 3.95157e-02
51 3.13792e-06 3.95157e-02
52 2.78126e-06 3.95157e-02
53 2.46578e-06 3.95156e-02
54 2.18679e-06 3.95156e-02
55 1.93936e-06 3.95156e-02
56 1.72059e-06 3.95156e-02
57 1.52675e-06 3.95156e-02
58 1.35470e-06 3.95156e-02
59 1.20293e-06 3.95156e-02
60 1.06741e-06 3.95156e-02
61 9.48208e-07 3.95156e-02
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.881 0.441 0.441]
cartesian multipoles:
[[-0.881 0. 0. 0.133 -5.381 -0. -0. -4.802 -0. -5.094 0. -0. 0.034 0. -0. 0. 0. 0.416 -0. 0.367]
[ 0.441 0. 0.022 -0.001 -0.275 -0. -0. -0.257 0.002 -0.26 0. 0.008 0.005 0. 0. 0. 0.024 0.009 0.009 0.023]
[ 0.441 0. -0.022 -0.001 -0.275 0. 0. -0.257 -0.002 -0.26 0. -0.008 0.005 0. -0. 0. -0.024 0.009 -0.009 0.023]]
Newton Method
The final method is the Newton method. Similarly to the previous methods, the Newton method cannot guarantee non-negative parameters, and negative pro-atom densities may arise during optimization. To address this, one might need to modify the hyper-parameters used in the method for different systems, which can make it less robust.
[6]:
def newton_method():
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = "newton"
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
newton_method()
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : NEWTON
Basis function type : GAUSS
Grid type : 1
Load GAUSS basis functions
Iteration Change Entropy
1 1.22855e-01 2.11396e-01
2 2.65514e-02 1.08102e-01
3 1.49501e-02 7.80048e-02
4 9.43137e-03 6.28961e-02
5 6.28278e-03 5.43378e-02
6 4.36240e-03 4.91980e-02
7 3.13687e-03 4.59888e-02
8 2.32436e-03 4.39260e-02
9 1.76689e-03 4.25692e-02
10 1.37223e-03 4.16592e-02
11 1.08483e-03 4.10387e-02
12 8.70228e-04 4.06094e-02
13 7.06492e-04 4.03086e-02
14 5.79250e-04 4.00953e-02
15 4.78831e-04 3.99426e-02
16 3.98556e-04 3.98323e-02
17 3.33687e-04 3.97519e-02
18 2.80794e-04 3.96929e-02
19 2.37332e-04 3.96492e-02
20 2.01386e-04 3.96168e-02
21 1.71486e-04 3.95926e-02
22 1.46492e-04 3.95743e-02
23 1.25506e-04 3.95606e-02
24 1.07816e-04 3.95501e-02
25 9.28500e-05 3.95422e-02
26 8.01470e-05 3.95361e-02
27 6.93317e-05 3.95315e-02
28 6.00974e-05 3.95279e-02
29 5.21920e-05 3.95251e-02
30 4.54072e-05 3.95230e-02
31 3.95702e-05 3.95214e-02
32 3.45374e-05 3.95201e-02
33 3.01887e-05 3.95191e-02
34 2.64235e-05 3.95183e-02
35 2.31572e-05 3.95177e-02
36 2.03187e-05 3.95173e-02
37 1.78476e-05 3.95169e-02
38 1.56930e-05 3.95166e-02
39 1.38115e-05 3.95164e-02
40 1.21661e-05 3.95162e-02
41 1.07252e-05 3.95161e-02
42 9.46198e-06 3.95160e-02
43 8.35316e-06 3.95159e-02
44 7.37883e-06 3.95158e-02
45 6.52185e-06 3.95158e-02
46 5.76738e-06 3.95158e-02
47 5.10260e-06 3.95157e-02
48 4.51639e-06 3.95157e-02
49 3.99910e-06 3.95157e-02
50 3.54233e-06 3.95157e-02
51 3.13875e-06 3.95157e-02
52 2.78197e-06 3.95157e-02
53 2.46641e-06 3.95156e-02
54 2.18717e-06 3.95156e-02
55 1.93998e-06 3.95156e-02
56 1.72107e-06 3.95156e-02
57 1.52714e-06 3.95156e-02
58 1.35528e-06 3.95156e-02
59 1.20294e-06 3.95156e-02
60 1.06787e-06 3.95156e-02
61 9.48078e-07 3.95156e-02
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.881 0.441 0.441]
cartesian multipoles:
[[-0.881 0. -0. 0.133 -5.381 -0. -0. -4.802 -0. -5.094 0. -0. 0.034 0. -0. 0. -0. 0.416 -0. 0.367]
[ 0.441 0. 0.022 -0.001 -0.275 -0. -0. -0.257 0.002 -0.26 0. 0.008 0.005 0. 0. 0. 0.024 0.009 0.009 0.023]
[ 0.441 0. -0.022 -0.001 -0.275 0. 0. -0.257 -0.002 -0.26 0. -0.008 0.005 0. -0. 0. -0.024 0.009 -0.009 0.023]]
Customized Methods
One can also apply customized solvers. Next, we will implement and apply a modified version of the self-consistent method.
[7]:
def customize_self_consistent_solver(
bs_funcs,
rho,
propars,
points,
weights,
threshold,
logger,
density_cutoff,
negative_cutoff,
population_cutoff,
):
r"""
Optimize parameters for proatom density functions using a self-consistent (SC) method.
.. math::
N_{Ai} = \int \rho_A(r) \frac{\rho_{Ai}^0(r)}{\rho_A^0(r)} dr
Parameters
----------
bs_funcs : 2D np.ndarray
Basis functions array with shape (M, N), where 'M' is the number of basis functions
and 'N' is the number of grid points.
rho : 1D np.ndarray
Spherically-averaged atomic density as a function of radial distance, with shape (N,).
propars : 1D np.ndarray
Pro-atom parameters with shape (M). 'M' is the number of basis functions.
points : 1D np.ndarray
Radial coordinates of grid points, with shape (N,).
weights : 1D np.ndarray
Weights for integration, including the angular part (4πr²), with shape (N,).
threshold : float
Convergence threshold for the iterative process.
density_cutoff : float
Density values below this cutoff are considered invalid.
negative_cutoff : float
The value less than `negative_cutoff` is treated as negative.
population_cutoff : float
The criteria for the difference between the sum of propars and reference population.
Returns
-------
1D np.ndarray
Optimized proatom parameters.
Raises
------
RuntimeError
If the inner iteration does not converge.
"""
pro_shells = bs_funcs * propars[:, None]
pro_density = np.einsum("ij->j", pro_shells)
sick = (rho < density_cutoff) | (pro_density < density_cutoff)
ratio = np.divide(rho, pro_density, out=np.zeros_like(rho), where=~sick)
propars[:] = np.einsum("p,ip->i", weights, pro_shells * ratio)
return propars
[8]:
def customized_self_consistent_method():
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = customize_self_consistent_solver
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
customized_self_consistent_method()
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : customize_self_consistent_solver
Basis function type : GAUSS
Grid type : 1
Load GAUSS basis functions
Iteration Change Entropy
1 8.86585e-02 2.11396e-01
2 5.02846e-02 1.36292e-01
3 2.65819e-02 1.03225e-01
4 2.18078e-02 8.44885e-02
5 1.89744e-02 7.24750e-02
6 1.50523e-02 6.43082e-02
7 1.12722e-02 5.85582e-02
8 8.31795e-03 5.44026e-02
9 6.29088e-03 5.13346e-02
10 5.01125e-03 4.90284e-02
11 4.21355e-03 4.72676e-02
12 3.67168e-03 4.59039e-02
13 3.24896e-03 4.48339e-02
14 2.88401e-03 4.39841e-02
15 2.55799e-03 4.33014e-02
16 2.27038e-03 4.27469e-02
17 2.02574e-03 4.22920e-02
18 1.82754e-03 4.19153e-02
19 1.67554e-03 4.16006e-02
20 1.56527e-03 4.13357e-02
21 1.48897e-03 4.11110e-02
22 1.43743e-03 4.09193e-02
23 1.40179e-03 4.07546e-02
24 1.37477e-03 4.06125e-02
25 1.35103e-03 4.04893e-02
26 1.32709e-03 4.03819e-02
27 1.30090e-03 4.02881e-02
28 1.27147e-03 4.02058e-02
29 1.23852e-03 4.01334e-02
30 1.20219e-03 4.00694e-02
31 1.16288e-03 4.00129e-02
32 1.12112e-03 3.99628e-02
33 1.07749e-03 3.99183e-02
34 1.03254e-03 3.98787e-02
35 9.86801e-04 3.98433e-02
36 9.40751e-04 3.98118e-02
37 8.94802e-04 3.97836e-02
38 8.49310e-04 3.97584e-02
39 8.04574e-04 3.97357e-02
40 7.60835e-04 3.97154e-02
41 7.18289e-04 3.96972e-02
42 6.77089e-04 3.96807e-02
43 6.37347e-04 3.96660e-02
44 5.99147e-04 3.96526e-02
45 5.62541e-04 3.96406e-02
46 5.27562e-04 3.96298e-02
47 4.94220e-04 3.96200e-02
48 4.62510e-04 3.96111e-02
49 4.32414e-04 3.96031e-02
50 4.03903e-04 3.95958e-02
51 3.76940e-04 3.95893e-02
52 3.51482e-04 3.95833e-02
53 3.27478e-04 3.95779e-02
54 3.04877e-04 3.95729e-02
55 2.83623e-04 3.95685e-02
56 2.63661e-04 3.95644e-02
57 2.44931e-04 3.95607e-02
58 2.27377e-04 3.95573e-02
59 2.10941e-04 3.95542e-02
60 1.95567e-04 3.95514e-02
61 1.81200e-04 3.95488e-02
62 1.67785e-04 3.95464e-02
63 1.55270e-04 3.95443e-02
64 1.43606e-04 3.95423e-02
65 1.32743e-04 3.95405e-02
66 1.22634e-04 3.95388e-02
67 1.13236e-04 3.95373e-02
68 1.04505e-04 3.95359e-02
69 9.64010e-05 3.95346e-02
70 8.88855e-05 3.95334e-02
71 8.19220e-05 3.95323e-02
72 7.54758e-05 3.95313e-02
73 6.95141e-05 3.95303e-02
74 6.40062e-05 3.95294e-02
75 5.89229e-05 3.95286e-02
76 5.42367e-05 3.95279e-02
77 4.99218e-05 3.95272e-02
78 4.59540e-05 3.95266e-02
79 4.23103e-05 3.95260e-02
80 3.89693e-05 3.95254e-02
81 3.59108e-05 3.95249e-02
82 3.31159e-05 3.95244e-02
83 3.05666e-05 3.95239e-02
84 2.82461e-05 3.95235e-02
85 2.61385e-05 3.95231e-02
86 2.42286e-05 3.95228e-02
87 2.25020e-05 3.95224e-02
88 2.09451e-05 3.95221e-02
89 1.95448e-05 3.95218e-02
90 1.82883e-05 3.95215e-02
91 1.71637e-05 3.95212e-02
92 1.61591e-05 3.95210e-02
93 1.52635e-05 3.95207e-02
94 1.44660e-05 3.95205e-02
95 1.37564e-05 3.95203e-02
96 1.31250e-05 3.95201e-02
97 1.25625e-05 3.95199e-02
98 1.20605e-05 3.95197e-02
99 1.16111e-05 3.95195e-02
100 1.12071e-05 3.95194e-02
101 1.08421e-05 3.95192e-02
102 1.05104e-05 3.95191e-02
103 1.02067e-05 3.95189e-02
104 9.92682e-06 3.95188e-02
105 9.66679e-06 3.95187e-02
106 9.42338e-06 3.95185e-02
107 9.19381e-06 3.95184e-02
108 8.97575e-06 3.95183e-02
109 8.76728e-06 3.95182e-02
110 8.56679e-06 3.95181e-02
111 8.37299e-06 3.95180e-02
112 8.18483e-06 3.95179e-02
113 8.00146e-06 3.95178e-02
114 7.82222e-06 3.95177e-02
115 7.64661e-06 3.95177e-02
116 7.47422e-06 3.95176e-02
117 7.30476e-06 3.95175e-02
118 7.13802e-06 3.95174e-02
119 6.97384e-06 3.95174e-02
120 6.81214e-06 3.95173e-02
121 6.65285e-06 3.95172e-02
122 6.49594e-06 3.95172e-02
123 6.34142e-06 3.95171e-02
124 6.18929e-06 3.95171e-02
125 6.03959e-06 3.95170e-02
126 5.89234e-06 3.95170e-02
127 5.74758e-06 3.95169e-02
128 5.60535e-06 3.95169e-02
129 5.46569e-06 3.95168e-02
130 5.32863e-06 3.95168e-02
131 5.19421e-06 3.95167e-02
132 5.06245e-06 3.95167e-02
133 4.93338e-06 3.95167e-02
134 4.80702e-06 3.95166e-02
135 4.68338e-06 3.95166e-02
136 4.56248e-06 3.95166e-02
137 4.44430e-06 3.95165e-02
138 4.32886e-06 3.95165e-02
139 4.21614e-06 3.95165e-02
140 4.10613e-06 3.95164e-02
141 3.99882e-06 3.95164e-02
142 3.89418e-06 3.95164e-02
143 3.79219e-06 3.95164e-02
144 3.69281e-06 3.95163e-02
145 3.59603e-06 3.95163e-02
146 3.50180e-06 3.95163e-02
147 3.41008e-06 3.95163e-02
148 3.32084e-06 3.95162e-02
149 3.23403e-06 3.95162e-02
150 3.14960e-06 3.95162e-02
151 3.06752e-06 3.95162e-02
152 2.98774e-06 3.95162e-02
153 2.91019e-06 3.95161e-02
154 2.83485e-06 3.95161e-02
155 2.76166e-06 3.95161e-02
156 2.69056e-06 3.95161e-02
157 2.62150e-06 3.95161e-02
158 2.55445e-06 3.95160e-02
159 2.48933e-06 3.95160e-02
160 2.42611e-06 3.95160e-02
161 2.36474e-06 3.95160e-02
162 2.30516e-06 3.95160e-02
163 2.24732e-06 3.95160e-02
164 2.19117e-06 3.95160e-02
165 2.13667e-06 3.95160e-02
166 2.08377e-06 3.95159e-02
167 2.03242e-06 3.95159e-02
168 1.98258e-06 3.95159e-02
169 1.93419e-06 3.95159e-02
170 1.88722e-06 3.95159e-02
171 1.84162e-06 3.95159e-02
172 1.79735e-06 3.95159e-02
173 1.75437e-06 3.95159e-02
174 1.71264e-06 3.95159e-02
175 1.67212e-06 3.95159e-02
176 1.63276e-06 3.95158e-02
177 1.59454e-06 3.95158e-02
178 1.55741e-06 3.95158e-02
179 1.52135e-06 3.95158e-02
180 1.48630e-06 3.95158e-02
181 1.45226e-06 3.95158e-02
182 1.41917e-06 3.95158e-02
183 1.38701e-06 3.95158e-02
184 1.35575e-06 3.95158e-02
185 1.32537e-06 3.95158e-02
186 1.29582e-06 3.95158e-02
187 1.26708e-06 3.95158e-02
188 1.23914e-06 3.95158e-02
189 1.21195e-06 3.95158e-02
190 1.18550e-06 3.95158e-02
191 1.15976e-06 3.95158e-02
192 1.13471e-06 3.95158e-02
193 1.11033e-06 3.95157e-02
194 1.08660e-06 3.95157e-02
195 1.06348e-06 3.95157e-02
196 1.04098e-06 3.95157e-02
197 1.01905e-06 3.95157e-02
198 9.97697e-07 3.95157e-02
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.881 0.441 0.441]
cartesian multipoles:
[[-0.881 -0. -0. 0.133 -5.381 -0. -0. -4.801 -0. -5.094 0. -0. 0.033 0. 0. 0. -0. 0.416 -0. 0.366]
[ 0.441 0. 0.022 -0.001 -0.275 -0. -0. -0.257 0.002 -0.26 0. 0.008 0.005 0. 0. 0. 0.024 0.009 0.009 0.023]
[ 0.441 0. -0.022 -0.001 -0.275 0. 0. -0.257 -0.002 -0.26 0. -0.008 0.005 0. -0. 0. -0.024 0.009 -0.009 0.023]]
One can also apply customized basis functions using the ExpBasisFuncHelper class. First, we need to define the basis functions, which should adhere to the following general format:
\[f_{ak} = c_{ak} \exp^{-\alpha_{ak} |r|^{n_{ak}}}\]
Here, \(c_{ak}\) represents the prefactors and the pro-atom parameters to be determined; \(\alpha_{ak}\) is the exponential coefficient; and \(n_{ak}\) corresponds to the power of the exponential function. The subscripts \(a\) and \(k\) denote the indices of atoms and basis functions, respectively.
We initialize ExpBasisFuncHelper using the default method, and therefore need to provide the \(\alpha_{ak}\) (exponents), \(n_{ak}\) (exponents_orders), and initial values of \(c_{ak}\) (initials). To illustrate the functionality, we use three exponential functions with different \(n\) values, i.e., 1 (Slater type), 1.5, and 2 (Gaussian type).
[9]:
bs_dict = {
"exponents_orders": {1: [1, 1.5, 2], 8: [1, 1.5, 2]},
"exponents": {1: [0.1, 0.2, 0.3], 8: [1.0, 2.0, 3.0]},
"initials": {1: [0.33] * 3, 8: [0.33] * 3},
}
basis_helper = ExpBasisFuncHelper(**bs_dict)
It should be noted that one can also initialize a ExpBasisFuncHelper through a json file by calling ExpBasisFuncHelper.from_json('json_file_path.json').
Next, we can apply the newly customized basis functions and the self-consistent solver. It is important to note
[10]:
def customized_basis_functions():
kwargs = prepare_argument_dict(mol, grid, rho)
kwargs["solver"] = customize_self_consistent_solver
kwargs["basis_func"] = basis_helper
part = LinearISAWPart(**kwargs)
part.do_all()
print_results(part)
customized_basis_functions()
================================================================================
Information of integral grids.
--------------------------------------------------------------------------------
Grid size of molecular grid: 18460
************************************ Atom 0 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
18 18 18 18 18 18 26 26 26 26 38 38 38 38 38 50 50 50 50 50
86 86 110 110 110 110 170 194 194 434 590 590 434 434 434 302 302 302 194 194
170 110 110 110 110 110 110 110 110 110 110 110 86 50 50 18 18 18 18 18
************************************ Atom 1 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
************************************ Atom 2 ************************************
|-- Radial grid size: 120
|-- Angular grid sizes:
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 18 18 18 18
18 18 18 18 18 18 18 18 18 26 26 26 38 38 38 38 38 38 38 50
50 50 50 50 50 86 86 86 110 110 110 170 170 170 194 194 194 194 170 170
170 110 110 110 110 110 110 110 110 110 110 86 86 50 50 50 18 18 18 18
--------------------------------------------------------------------------------
================================================================================
Performing a density-based AIM analysis with a wavefunction as input.
Molecular grid : MolGrid
Using local grids : True
Scheme : Linear Iterative Stockholder
Outer loop convergence threshold : 1.0e-06
Inner loop convergence threshold : 1.0e-08
Using global ISA : False
Maximum outer iterations : 1000
lmax : 3
Solver : customize_self_consistent_solver
Basis function type : Customized
Grid type : 1
Iteration Change Entropy
1 5.89735e-01 5.05801e+00
2 1.15038e-01 3.85235e+00
3 3.22005e-02 3.65738e+00
4 2.93260e-02 3.54683e+00
5 3.14458e-02 3.47062e+00
6 3.00283e-02 3.41504e+00
7 2.73979e-02 3.37363e+00
8 2.46431e-02 3.34244e+00
9 2.21092e-02 3.31880e+00
10 1.98673e-02 3.30078e+00
11 1.79024e-02 3.28696e+00
12 1.61789e-02 3.27630e+00
13 1.46612e-02 3.26803e+00
14 1.33189e-02 3.26156e+00
15 1.21268e-02 3.25646e+00
16 1.10643e-02 3.25241e+00
17 1.01144e-02 3.24917e+00
18 9.26283e-03 3.24657e+00
19 8.49748e-03 3.24445e+00
20 7.80804e-03 3.24272e+00
21 7.18563e-03 3.24130e+00
22 6.62257e-03 3.24013e+00
23 6.11219e-03 3.23915e+00
24 5.64868e-03 3.23833e+00
25 5.22694e-03 3.23764e+00
26 4.84253e-03 3.23706e+00
27 4.49154e-03 3.23656e+00
28 4.17053e-03 3.23614e+00
29 3.87647e-03 3.23578e+00
30 3.60667e-03 3.23548e+00
31 3.35878e-03 3.23521e+00
32 3.13068e-03 3.23498e+00
33 2.92052e-03 3.23478e+00
34 2.72662e-03 3.23461e+00
35 2.54751e-03 3.23446e+00
36 2.38186e-03 3.23433e+00
37 2.22849e-03 3.23422e+00
38 2.08632e-03 3.23412e+00
39 1.95441e-03 3.23403e+00
40 1.83189e-03 3.23395e+00
41 1.71799e-03 3.23389e+00
42 1.61201e-03 3.23383e+00
43 1.51330e-03 3.23378e+00
44 1.42129e-03 3.23373e+00
45 1.33547e-03 3.23369e+00
46 1.25535e-03 3.23366e+00
47 1.18051e-03 3.23363e+00
48 1.11055e-03 3.23360e+00
49 1.04510e-03 3.23358e+00
50 9.83842e-04 3.23356e+00
51 9.26471e-04 3.23354e+00
52 8.72711e-04 3.23352e+00
53 8.22307e-04 3.23351e+00
54 7.75025e-04 3.23349e+00
55 7.30651e-04 3.23348e+00
56 6.88987e-04 3.23347e+00
57 6.49850e-04 3.23346e+00
58 6.13072e-04 3.23345e+00
59 5.78496e-04 3.23345e+00
60 5.45978e-04 3.23344e+00
61 5.15385e-04 3.23344e+00
62 4.86593e-04 3.23343e+00
63 4.59488e-04 3.23343e+00
64 4.33962e-04 3.23342e+00
65 4.09916e-04 3.23342e+00
66 3.87259e-04 3.23342e+00
67 3.65904e-04 3.23341e+00
68 3.45771e-04 3.23341e+00
69 3.26786e-04 3.23341e+00
70 3.08880e-04 3.23341e+00
71 2.91987e-04 3.23341e+00
72 2.76047e-04 3.23341e+00
73 2.61003e-04 3.23340e+00
74 2.46802e-04 3.23340e+00
75 2.33394e-04 3.23340e+00
76 2.20734e-04 3.23340e+00
77 2.08777e-04 3.23340e+00
78 1.97482e-04 3.23340e+00
79 1.86813e-04 3.23340e+00
80 1.76731e-04 3.23340e+00
81 1.67205e-04 3.23340e+00
82 1.58202e-04 3.23340e+00
83 1.49692e-04 3.23340e+00
84 1.41648e-04 3.23340e+00
85 1.34043e-04 3.23340e+00
86 1.26852e-04 3.23340e+00
87 1.20053e-04 3.23340e+00
88 1.13623e-04 3.23340e+00
89 1.07543e-04 3.23340e+00
90 1.01791e-04 3.23340e+00
91 9.63514e-05 3.23340e+00
92 9.12053e-05 3.23340e+00
93 8.63370e-05 3.23340e+00
94 8.17312e-05 3.23340e+00
95 7.73735e-05 3.23340e+00
96 7.32502e-05 3.23340e+00
97 6.93485e-05 3.23340e+00
98 6.56563e-05 3.23340e+00
99 6.21623e-05 3.23340e+00
100 5.88555e-05 3.23340e+00
101 5.57259e-05 3.23340e+00
102 5.27638e-05 3.23340e+00
103 4.99602e-05 3.23340e+00
104 4.73064e-05 3.23340e+00
105 4.47943e-05 3.23340e+00
106 4.24164e-05 3.23340e+00
107 4.01653e-05 3.23340e+00
108 3.80343e-05 3.23340e+00
109 3.60168e-05 3.23340e+00
110 3.41069e-05 3.23340e+00
111 3.22986e-05 3.23340e+00
112 3.05866e-05 3.23340e+00
113 2.89656e-05 3.23340e+00
114 2.74309e-05 3.23340e+00
115 2.59777e-05 3.23340e+00
116 2.46018e-05 3.23340e+00
117 2.32989e-05 3.23340e+00
118 2.20653e-05 3.23340e+00
119 2.08971e-05 3.23340e+00
120 1.97909e-05 3.23340e+00
121 1.87435e-05 3.23340e+00
122 1.77516e-05 3.23340e+00
123 1.68123e-05 3.23340e+00
124 1.59228e-05 3.23340e+00
125 1.50804e-05 3.23340e+00
126 1.42827e-05 3.23340e+00
127 1.35273e-05 3.23340e+00
128 1.28119e-05 3.23340e+00
129 1.21343e-05 3.23340e+00
130 1.14927e-05 3.23340e+00
131 1.08850e-05 3.23340e+00
132 1.03095e-05 3.23340e+00
133 9.76452e-06 3.23340e+00
134 9.24834e-06 3.23340e+00
135 8.75948e-06 3.23340e+00
136 8.29649e-06 3.23340e+00
137 7.85799e-06 3.23340e+00
138 7.44269e-06 3.23340e+00
139 7.04936e-06 3.23340e+00
140 6.67684e-06 3.23340e+00
141 6.32402e-06 3.23340e+00
142 5.98985e-06 3.23340e+00
143 5.67336e-06 3.23340e+00
144 5.37360e-06 3.23340e+00
145 5.08968e-06 3.23340e+00
146 4.82078e-06 3.23340e+00
147 4.56610e-06 3.23340e+00
148 4.32487e-06 3.23340e+00
149 4.09640e-06 3.23340e+00
150 3.88000e-06 3.23340e+00
151 3.67504e-06 3.23340e+00
152 3.48091e-06 3.23340e+00
153 3.29704e-06 3.23340e+00
154 3.12289e-06 3.23340e+00
155 2.95794e-06 3.23340e+00
156 2.80170e-06 3.23340e+00
157 2.65372e-06 3.23340e+00
158 2.51356e-06 3.23340e+00
159 2.38080e-06 3.23340e+00
160 2.25506e-06 3.23340e+00
161 2.13596e-06 3.23340e+00
162 2.02315e-06 3.23340e+00
163 1.91630e-06 3.23340e+00
164 1.81510e-06 3.23340e+00
165 1.71924e-06 3.23340e+00
166 1.62845e-06 3.23340e+00
167 1.54245e-06 3.23340e+00
168 1.46099e-06 3.23340e+00
169 1.38384e-06 3.23340e+00
170 1.31076e-06 3.23340e+00
171 1.24154e-06 3.23340e+00
172 1.17597e-06 3.23340e+00
173 1.11387e-06 3.23340e+00
174 1.05505e-06 3.23340e+00
175 9.99340e-07 3.23340e+00
Computing atomic populations.
Computing atomic charges.
Computing density decomposition for atom 0
Computing density decomposition for atom 1
Computing density decomposition for atom 2
Computing cartesian and pure AIM multipoles and radial AIM moments.
Storing proatom density spline for atom 0.
Storing proatom density spline for atom 1.
Storing proatom density spline for atom 2.
charges:
[-0.411 0.207 0.207]
cartesian multipoles:
[[-0.411 0. -0. 0.05 -3.886 -0. -0. -3.749 -0. -4.009 -0. -0. -0.561 0. 0. -0. -0. 0.281 -0. -0.98 ]
[ 0.207 -0. 0.243 -0.22 -1.022 -0. -0. -0.931 0.077 -1.001 -0. 0.487 -0.535 -0. 0. -0. 1.249 -0.268 0.399 -1.297]
[ 0.207 -0. -0.243 -0.22 -1.022 -0. -0. -0.931 -0.077 -1.001 -0. -0.487 -0.535 -0. 0. -0. -1.249 -0.268 -0.399 -1.297]]
[ ]: