First Principles Calculation of Optically-Excited Atomic Forces

Sohrab Ismail-Beigi and Steven G. Louie,
Department of Physics, University of California at Berkeley, and
Lawrence Berkeley National Laboratory

Motivation --- When optically excited, many materials emit light, or luminesce, with spectrum that is different than that which was absorbed. This effect is quite general and occurs in molecules or the solid state and has its microscopic origin in the fact that when photon absorption leads to excitation of electrons, the resulting change in the electronic wavefunction can cause the material's constituent atoms to rearrange, thus changing the optical properties of the material. Theoretical understanding such photo-induced structural change addresses not only important scientific questions about the relation of optical properties to microscopic atomic structure, but also is of technological significance in the engineering and design of materials with desired optical features.

Recently, calculation of optically excited electronic properties has become feasible within the first-principles GW-Bethe Salpeter equation (GW-BSE) formalism [1]. However, the current methodology only provides optical absorption spectra at fixed atomic positions: without using either shrewd guess or an inefficient finite-difference scheme, there is no way to know in which direction the excited atoms will move to optimize the geometry. Knowledge of the excited-state forces on the atoms would allow for calculation of the relaxations, the luminescence spectrum, and the possibility of excited-state molecular dynamics.

Accomplishments --- In the past year, we have completed our work on the gauge-invariant coupling of electromagnetic fields to nonlocal potentials through publication [2]. In addition, we have expanded the study to include the role of nonlocality in coupling tight-binding or hopping Hamiltonians to electromagnetic fields as well as the proper form of coupling for effective-mass theories [3].

Regarding excited-state forces, we have completed the derivation of a new formalism for the efficient calculation of the excited-state forces within the GW-BSE methodology. This required a careful consideration of the importance of various contributions and the use of appropriate, physically motivated approximations in key places to allow for tractable calculation. As a proof of principle and accuracy of the GW-BSE method and the resulting forces, we have studied the relaxation of photo-excited molecular carbon dioxide and ammonia, both of which have been studied in detail experimentally. We find that the GW-BSE method provides excellent for the excited state properties and that our resulting forces are accurate for efficient relaxation of the atomic geometries [4].

Click on figure icon to the left for a figure showing the dependence of the ground-state and excited-state energy for carbon monoxide on the bond length R (can be viewed at http://civet.berkeley.edu/sohrab/cmsn/plotebsevsd.pdf).


The table below shows a comparison of the ground- and excited-state properties as calculated using our theoretical methods to available experimental data for carbon monoxide. We have included results from the constrained LDA (CLDA) method as well. IP stands for the ionization potential; w is the vibrational frequency of the bond; and Te is the transition energy (minimum-to-minimum).

Carbon monoxide
Ground stateR (Å)w (cm-1)IP (eV)
LDA1.13 2050 9.1
GW--- --- 14.1
Experiment 1.1283 2169.8 14.01
First excited stateR (Å)w (cm-1)Te (eV)
CLDA1.21 1720 7.02
GW-BSE1.26 1290 8.32
Experiment 1.235 1518 8.07

For ammonia, it is well known that while the ground state of the molecule is pyramidal (experimental bond length 1.01 Å and bond angle 106.7o compared to the LDA bond length 1.03 Å and bond angle 105.0o), the first excited state has a flat, planar geometry. Our GW-BSE forces relax ammonia into a planar geometry and with a bond length of 1.08 Å (same as in experiment); the GW-BSE transition energy (Te) is 5.52 eV as compared to the experimental value of 5.7 eV.

In addition to these findings, we have checked that the forces we calculate within our formalism are indeed the derivatives of the energy of the excited state. We find that forces are typically given to better than a few hundredths of eV/Å which leads to typical errors in bond lengths of order 0.02 Å or smaller.

Significance --- Our work provides a new formalism for calculating the forces on atoms in optically excited states within the GW-BSE methodology. We have implemented and tested the accuracy of the formalism by studying the carbon dioxide and ammonia molecules and found excellent agreement with experiment as well as precise forces. The ability to compute excited-state forces opens new doors in the study of optical properties of materials: one can begin to address previously computationally inaccessible questions regarding photo-luminescence spectra, relaxation of atomic structure after absorption of light, or the microscopic structure of photo-induced defects such as self-trapped excitons.

References:

  1. M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 81 2312 (1998); S. Albrecht, L. Reining, R. Del Sole, and G Onida, Phys. Rev. Lett. 80 4510 (1998); L. X. Benedict, E. L. Shirley, and R. B. Bohn, Phys. Rev. Lett. 80, 4514 (1998); M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000).
  2. S. Ismail-Beigi, E. K. Chang, and S. G. Louie, Phys. Rev. Lett. 87 087402 (2001). Available electronically at http://arXiv.org/abs/cond-mat/0101383.
  3. Invited talk at the The Workshop on Recent Developments in Electronic Structure Methods, Princeton, NJ, June 2001; publication by S. Ismail-Beigi and S. G. Louie in preparation.
  4. Currently under review at Phys. Rev. Lett. Preprint is available at http://xxx.lanl.gov/abs/cond-mat/0207248.

Contact:

S. G. Louie, Univ. of California, Berkeley, CA, 94720, Phone: (510) 642-1709, Fax: (510) 643-9473, email: sglouie@uclink4.berkeley.edu