# MP2 polarizability using finite field

Hello, I would like to use MP2 dipole moments to obtain MP2 dipole polarizability based on the finite-field approach. I tried two different methods to obtain MP2 relaxed dipole moments under an electric field but did not work. Any suggestions? Thanks.

(1)
pert = 1.8897261250e-5
set perturb_h true
set perturb_with dipole
set perturb_dipole [pert, 0, 0]
properties(‘mp2’, properties=[‘dipole’])
dpx = psi4.get_variable(‘MP2 DIPOLE X’)

(2)
pert = 1.8897261250e-5
scf_e, scf_wfn = energy(‘scf’, return_wfn=True)
mints = psi4.core.MintsHelper(scf_wfn)
dipole = mints.ao_dipole()
dipole[0].scale(pert) # dipole_x perturbation
Perturb_matrix = scf_wfn.Fa()
Perturb_matrix.add(dipole[0])
properties(‘mp2’,properties=[‘dipole’],ref_wfn=scf_wfn)
dpx = psi4.get_variable(‘MP2 DIPOLE X’)

I modified the finite difference dipole test as follows:

``````molecule {
symmetry c1 # this is important; some perturbations will break symmetry
no_reorient
O          -0.947809457408    -0.132934425181     0.000000000000
H          -1.513924046286     1.610489987673     0.000000000000
F           0.878279174340     0.026485523618     0.000000000000
}

pert = 0.001
lambdas = [pert, -pert, 2.0*pert, -2.0*pert]

set {
basis     cc-pVDZ
d_convergence   10
}
method = 'MP2'

perturbed_energies = np.zeros((len(lambdas),3))
x_perturbed_dipoles = np.zeros((len(lambdas),3))
y_perturbed_dipoles = np.zeros((len(lambdas),3))
z_perturbed_dipoles = np.zeros((len(lambdas),3))

# start with a reference dipole calculation
properties(method, properties=['dipole'])
mu_x = psi4.core.variable(method + ' DIPOLE X') / psi_dipmom_au2debye
mu_y = psi4.core.variable(method + ' DIPOLE Y') / psi_dipmom_au2debye
mu_z = psi4.core.variable(method + ' DIPOLE Z') / psi_dipmom_au2debye
analytic_dipole = np.array([mu_x, mu_y, mu_z])

# now compute with different applied fields
for step, l in enumerate(lambdas):
set perturb_h true
set perturb_with dipole
# x pertubation
set perturb_dipole [\$l, 0, 0]
perturbed_energies[step,0] = properties(method, properties=['dipole'])
x_perturbed_dipoles[step,0] = psi4.core.variable(method + ' DIPOLE X')
x_perturbed_dipoles[step,1] = psi4.core.variable(method + ' DIPOLE Y')
x_perturbed_dipoles[step,2] = psi4.core.variable(method + ' DIPOLE Z')
# y pertubation
set perturb_dipole [0, \$l, 0]
perturbed_energies[step,1] = properties(method, properties=['dipole'])
y_perturbed_dipoles[step,0] = psi4.core.variable(method + ' DIPOLE X')
y_perturbed_dipoles[step,1] = psi4.core.variable(method + ' DIPOLE Y')
y_perturbed_dipoles[step,2] = psi4.core.variable(method + ' DIPOLE Z')
# z pertubation
set perturb_dipole [0, 0, \$l]
perturbed_energies[step,2] = properties(method, properties=['dipole'])
z_perturbed_dipoles[step,0] = psi4.core.variable(method + ' DIPOLE X')
z_perturbed_dipoles[step,1] = psi4.core.variable(method + ' DIPOLE Y')
z_perturbed_dipoles[step,2] = psi4.core.variable(method + ' DIPOLE Z')

# use 3- and 5-point finite difference formulae to compute the dipole
mu_3pt = (perturbed_energies[0] - perturbed_energies[1]) / (2.0*pert)
mu_5pt = (8.0*perturbed_energies[0] - 8.0*perturbed_energies[1] - perturbed_energies[2] + perturbed_energies[3]) / (12.0*pert)
# and the polarizability
a_x_3pt = (x_perturbed_dipoles[0] - x_perturbed_dipoles[1]) / (2.0*pert)
a_y_3pt = (y_perturbed_dipoles[0] - y_perturbed_dipoles[1]) / (2.0*pert)
a_z_3pt = (z_perturbed_dipoles[0] - z_perturbed_dipoles[1]) / (2.0*pert)
alpha_3pt = np.vstack((a_x_3pt, a_y_3pt, a_z_3pt))
a_x_5pt = (8.0*x_perturbed_dipoles[0] - 8.0*x_perturbed_dipoles[1] - x_perturbed_dipoles[2] + x_perturbed_dipoles[3]) / (12.0*pert)
a_y_5pt = (8.0*y_perturbed_dipoles[0] - 8.0*y_perturbed_dipoles[1] - y_perturbed_dipoles[2] + y_perturbed_dipoles[3]) / (12.0*pert)
a_z_5pt = (8.0*z_perturbed_dipoles[0] - 8.0*z_perturbed_dipoles[1] - z_perturbed_dipoles[2] + z_perturbed_dipoles[3]) / (12.0*pert)
alpha_5pt = np.vstack((a_x_5pt, a_y_5pt, a_z_5pt))

# let's see what happened!
np.set_printoptions(suppress=True)
print('Analytic dipole\n', analytic_dipole)
print('3point finite difference dipole\n', mu_3pt)
print('5point finite difference dipole\n', mu_5pt)

print('3point finite difference polarizability\n', alpha_3pt)
print('5point finite difference polarizability\n', alpha_5pt)
``````

which generates the following results:

``````Analytic dipole
[-0.03758964  0.36627481  0.        ]
3point finite difference dipole
[-0.03779228  0.36633549  0.        ]
5point finite difference dipole
[-0.03758969  0.36627479  0.        ]
3point finite difference polarizability
[[-42.5552801   22.72507235   0.        ]
[ 22.72614401 -59.60566524   0.        ]
[  0.00000001   0.00000001  -6.65858834]]
5point finite difference polarizability
[[-42.55982767  22.73135971   0.        ]
[ 22.73135706 -59.61113539   0.        ]
[  0.00000001   0.00000001  -6.65858666]]
``````

Assuming I didn’t mess up somewhere, it appears that the 3 point finite difference approach is pretty good and can probably be used instead of the 5 point stencil, saving a number of computations. The tensor is not symmetrized, so the degree of asymmetry is a good indicator of the noise in the procedure, which looks quite reasonable in this simple test case.

Thank you! It is working now.^^