# [DFTB-Plus-User] DFTB+ about the process for calculating relaxed density

이인성 (화학과) islee at unist.ac.kr
Wed May 31 05:45:43 CEST 2017

Thank you for answering my question.

As you mentioned, the ground state dipole is equal to the multiplication of mulliken charge and coordinates of atoms.

This means that   dipole moment = - \partial(E) / \partial(F) = charge * coordinates

But, what I want to know is that dipole moment integrals which is expressed by    d_mu_nu = e * < chi_mu | r | chi_nu > = \partial(h_mu_nu) / \partial(F)

Here, d_mu_nu is dipole moment integrals in the AO basis, h_mu_nu is hamiltonian (it may include only one-electron part).

When we calculate the dipole moment without using directly charges and coordinates, the only non-vanishing derivatives of the molecular integrals are the dipole moment integrals. so I want to know the analytic (exact) form of this dipole moment integrals.

As you said, I finally want to calculate electronic excitations.

Thank you.

In Seong Lee

________________________________
보낸 사람: Ben Hourahine <benjamin.hourahine at strath.ac.uk> 대신 DFTB-Plus-User <dftb-plus-user-bounces at mailman.zfn.uni-bremen.de>
보낸 날짜: 2017년 5월 30일 화요일 오후 7:26:19
받는 사람: dftb-plus-user at mailman.zfn.uni-bremen.de
제목: Re: [DFTB-Plus-User] DFTB+ about the process for calculating relaxed density

Dear Seong Lee,

The ground state dipole is, as you mention, the derivative of energy with respect to the field. This can be calculated using the Hellmann-Feynman theorem through the derivative of the hamiltonian with respect to the external field. This gives the same value as obtained using the Mulliken charges and locations of atoms (this is due to the use of Mulliken partitioning in constructing the contribution from an external electric fields to the total energy).

When DFTB+ is compiled at DEBUG >= 2, it will evaluate dipole moments for molecules using both of these methods.

The density matrix is written in the basis of the atomic orbitals used for DFTB and it is these functions that are explicitly dependent on r, not the density matrix itself. DFTB avoids evaluation of integrals during the calculation by re-casting as much as possible in terms of the overlap elements between atomic basis functions. So for example, the contribution from an external field is

H_ij = 0.5 S_ij ( V_i + V_j)

where V_i is the external potential at the atom containing orbital i. There is more detail in several of the articles listed at http://www.dftb.org/about-dftb/references/

Are you instead asking about the oscillator strength for something else, such as electronic or vibrational excitations?

Regards

Ben

On 30/05/17 10:33, 이인성 (화학과) wrote:

Hello, DFTB+ developers

I am a new user for dftb+ and I want to know about relaxed density matrix.

To obtain the information about the oscillator strength, we have to know relaxed density matrix.

And to obtain the relaxed density matrix, we have to know the dipole moment integrals.

Differentiation of the total energy with respect to the field F gives the dipole moment.

d_mu_nu = e * < chi_mu | r | chi_nu > = \partial(h_mu_nu) / \partial(field)

where d_mu_nu is dipole moment integrals, chi is AO, e is electron charge, and r is the electron coordinate vector.

I am sorry for low quality of upper equations.

Thus, I want to know 1. the analytic formula for d_mu_nu and 2. the process how dftb+ calculate this dipole moment for oscillator strength.

Thank you.

In Seong Lee

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