Finite-field Coupled Cluster calculations of dipole moment and polarizability

BeH electric properties

This toy molecule in the small STO-2G decontracted basis demonstrates the use of finite-field perturbation theory for calculating the dipole moment (\(\mu_{z}\)) and the polarizability (\(\alpha_{zz}\)) both at the Hartree-Fock and at the highly correlated Coupled Cluster level. Our molecule is oriented in the z-axis.

Because BeH is open-shell system, we can not employ MP2 and CCSD gradients, which are available only for closed-shell systems. We have to resort to the finite-field perturbation scheme, where we add the z-dipole moment operator as small perturbation to the X2C relativistic Hamiltonian. The symmetry of heterodiatomic molecule must be lowered from linear to C2v because of the z-oriented dipole perturbation operator.

Molecule in electric field

When a molecular system is placed in a homogenous static electric field, F=[Fx,Fy,Fz] (p,q-components), its energy may be expanded in a Taylor serie as follows:

(1)\[E=E_{0}-\mu_{p}F_{p} - \frac{1}{2}\alpha_{pq}F_{p}F_{q} ...\]

We keep only z-component of the field, F=Fz, and the dipole moment is obtained as the first numerical derivative of the energy according to the electric field at zero field strength:

(2)\[\mu_{z}=-\Biggl( \frac{\partial E(F_{z})}{\partial F_{z}} \Biggr)_{F_{z}=0}\]

The polarizability (for simplicity, here the zz-component) is calculated via second numerical derivative of the energy according to the electric field at zero field strength:

(3)\[\alpha_{zz}=-\Biggl( \frac{\partial^{2} E(F_{z})}{\partial F^{2}_{z}} \Biggr)_{F_{z}=0}\]

Method of computation

Let us first compute the total energies of few electric field strength. This can be done through replacing the string in the DIRAC input file by the field strength:

pam --noarch  --replace zff=+0.0005 --mol=BeH.sto-2g.C2v.mol --inp=BeH.x2c_scf_relcc_ffz.inp --put "DFPCMO.BeH.x2c_scf_sto-2g.C2v=DFPCMO"

(Input files for download are BeH.x2c_scf_relcc_ffz.inp, BeH.sto-2g.C2v.mol and the text MO file, DFPCMO.BeH.x2c_scf_sto-2g.C2v .)

The total SCF and CCSD(T) energies varrying on field strengths are sorted in the following Table:

Perturbed energies
Fz E(SCF,Fz) E(CCSD(T),Fz)
+0.0010 -14.424976681856371 -14.461324673372751
+0.0005 -14.425860788235255 -14.462193434302819
+0.0000 -14.426746572346620 -14.463063980838802
-0.0005 -14.427634036448282 -14.463936314225093
-0.0010 -14.428523182834038 -14.464810435757153

NOTE: In this quick DIRAC test, only two output files are provided for Coupled Cluster method, BeH.x2c_scf_relcc_-0.0005_BeH.sto-2g.C2v.out , BeH.x2c_scf_relcc_+0.0005_BeH.sto-2g.C2v.out.


In the attached spreadsheet (accesible via Gnumerics, Libre Office Calc or MS Excel), we interpolate five pairs, [F,E(F)], of the Table Perturbed energies with the 4th-degree polynomial:


The negative first derivative of the polynomial (4) at zero field is the dipole moment, based on the equation (2):

(5)\[\mu_{z} = - \Biggl( \frac{\partial E(F)}{\partial F} \Biggr)_{F=0} = -a_{1}\]

The second derivative (with minus sign) is the polarizability, following prescription in (3):

(6)\[\alpha_{zz} =- \Biggl( \frac{\partial^{2} E(F)}{\partial F^{2}} \Biggr)_{F=0} = -2a_{2}\]

In the spreadsheet environment we employ the function LINEST. For that you have to prepare columns with finite-field stregths powered to 1, 2, 3 and 4.

The binary spreadsheet file available for download is data.ods.

Dipole moment

We calculate here the electronic part of the z-component of the dipole moment as the perturbing field goes in the z-direction.

Expectation value

For the SCF method, we can obtain the dipole moment - both for closed and open-shell systems - via the expectation value, which gives all three components, \(\mu_{x}\), \(\mu_{y}\) and \(\mu_{z}\) . A good practise is that dipole moment finite-field calculations are verified against the SCF/DFT expectation value.

The SCF expectation value dipole moment - its zz-electronic contribution - is \(\mu_{z}\) =-1.77324745 au. (The input file for download, BeH.x2c_scf_dipmom.inp and the the corresponding output file, BeH.x2c_scf_dipmom_BeH.sto-2g.C2v.out.)

Finite-field values

From the spreadsheet function table, we obtain the value of \(a_{1}\) =-1.7732474 a.u., which is the expansion coefficient of the first order field.

Simple first numerical derivation of the SCF perturbed energy according to the applied electric field, equation (2),

(7)\[\mu_{z} = - \Biggl( \frac{E(+F)-E(-F)}{2F} \Biggr),\]

gives -1.771568 a.u. for F=0.0005. For the field strength of F=0.001 it is -1.76989 a.u.

The zero field derivative of the fourth order polynomial, equation (5) is more precise when comparing against the above mentioned SCF expectation value, though finite field evaluation is still capable to give value accurate to two decimal places.


For closed-shell systems, the polarizability at the SCF and DFT levels can be calculated via linear response method, which gives all components of the polarizability tensor. For systems with unpaired electrons like this one, however, we have to resort to finite-field perturbative calculations. For simplicity, we focus on the zz-tesnor component of the polarizability, \(\alpha_{zz}\)


For our molecule, the simple numerical second derivate calculation for given F=0.0005 at SCF level, equation (3)

(8)\[\alpha_{zz} = - \Biggl( \frac{E(+F)+E(-F)-2E(0)}{F^{2}} \Biggr),\]

gives 6.71996 a.u.. This is close to the second order expansion coefficient, multiplied by 2, giving the value of \(\alpha_{zz}=-2a_{2}\) =(-2)*(-3.3599745995852954)=6.71994919 a.u.


Similar numerical derivation for perturbed CCSD(T) energies and for F=0.0005, according to equation (8), gives the value of \(\alpha_{zz}\) =7.147401232 a.u. The corresponding expansion coefficient (multiplied by the factor of two) from equation (6), produces the value of \(\alpha_{zz}\) =7.14738420 a.u.

One can conclude that the F=0.0005 field strength is sufficient to obtain polarizability accurate to 3 decimal places by simple numerical second derivation, equation (8).