Nodal variational principle for excited states
Abstract
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. In conclusion,more »
- Authors:
-
- Ames Lab. and Iowa State Univ., Ames, IA (United States)
- Duke Univ., Durham, NC (United States); North Carolina A&T State Univ., Greensboro, NC (United States); Tulane Univ., New Orleans, LA (United States)
- Publication Date:
- Research Org.:
- Ames Lab., Ames, IA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1466358
- Alternate Identifier(s):
- OSTI ID: 1462305
- Report Number(s):
- IS-J-9729
Journal ID: ISSN 2469-9926; PLRAAN
- Grant/Contract Number:
- AC02-07CH11358
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review A
- Additional Journal Information:
- Journal Volume: 98; Journal Issue: 1; Journal ID: ISSN 2469-9926
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS
Citation Formats
Zahariev, Federico, Gordon, Mark S., and Levy, Mel. Nodal variational principle for excited states. United States: N. p., 2018.
Web. doi:10.1103/PhysRevA.98.012144.
Zahariev, Federico, Gordon, Mark S., & Levy, Mel. Nodal variational principle for excited states. United States. https://doi.org/10.1103/PhysRevA.98.012144
Zahariev, Federico, Gordon, Mark S., and Levy, Mel. Tue .
"Nodal variational principle for excited states". United States. https://doi.org/10.1103/PhysRevA.98.012144. https://www.osti.gov/servlets/purl/1466358.
@article{osti_1466358,
title = {Nodal variational principle for excited states},
author = {Zahariev, Federico and Gordon, Mark S. and Levy, Mel},
abstractNote = {It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. In conclusion, simple numerical illustrations of the error functions are presented.},
doi = {10.1103/PhysRevA.98.012144},
journal = {Physical Review A},
number = 1,
volume = 98,
place = {United States},
year = {Tue Jul 31 00:00:00 EDT 2018},
month = {Tue Jul 31 00:00:00 EDT 2018}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 4 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Vertical and adiabatic excitations in anthracene from quantum Monte Carlo: Constrained energy minimization for structural and electronic excited-state properties in the JAGP ansatz
journal, June 2015
- Dupuy, Nicolas; Bouaouli, Samira; Mauri, Francesco
- The Journal of Chemical Physics, Vol. 142, Issue 21
Symmetry, nodal surfaces, and energy ordering of molecular orbitals
journal, December 1975
- Wilson, E. Bright
- The Journal of Chemical Physics, Vol. 63, Issue 11
Incipient Wigner localization in circular quantum dots
journal, August 2007
- Ghosal, Amit; Güçlü, A. D.; Umrigar, C. J.
- Physical Review B, Vol. 76, Issue 8
Generalized variational principle for excited states using nodes of trial functions
journal, October 2011
- Bressanini, Dario; Reynolds, Peter J.
- Physical Review E, Vol. 84, Issue 4
Numerische Berechnung der 2S-Terme von Ortho- und Par-Helium
journal, November 1930
- Hylleraas, Egil A.; Undheim, Bjarne
- Zeitschrift f�r Physik, Vol. 65, Issue 11-12
Nodes of entangled -particle wave functions
journal, December 2006
- Briggs, John S.; Walter, Michael
- Physical Review A, Vol. 74, Issue 6
Towards a systematic assessment of errors in diffusion Monte Carlo calculations of semiconductors: Case study of zinc selenide and zinc oxide
journal, December 2015
- Yu, Jaehyung; Wagner, Lucas K.; Ertekin, Elif
- The Journal of Chemical Physics, Vol. 143, Issue 22
Quantized vortices around wavefunction nodes. II
journal, December 1974
- Hirschfelder, Joseph O.; Goebel, Charles J.; Bruch, Ludwig W.
- The Journal of Chemical Physics, Vol. 61, Issue 12
Quantum Monte Carlo calculations of the optical gaps of Ge nanoclusters using core-polarization potentials
journal, January 2007
- Vincent, Jordan E.; Kim, Jeongnim; Martin, Richard M.
- Physical Review B, Vol. 75, Issue 4
Introducing QMC/MMpol: Quantum Monte Carlo in Polarizable Force Fields for Excited States
journal, March 2016
- Guareschi, Riccardo; Zulfikri, Habiburrahman; Daday, Csaba
- Journal of Chemical Theory and Computation, Vol. 12, Issue 4
Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states
journal, May 2011
- Hipes, Paul G.
- Physical Review B, Vol. 83, Issue 19
Nodal surfaces of helium atom eigenfunctions
journal, June 2007
- Scott, Tony C.; Lüchow, Arne; Bressanini, Dario
- Physical Review A, Vol. 75, Issue 6
Communication: Spectroscopic consequences of proton delocalization in OCHCO +
journal, August 2015
- Fortenberry, Ryan C.; Yu, Qi; Mancini, John S.
- The Journal of Chemical Physics, Vol. 143, Issue 7
Fixed‐node quantum Monte Carlo for molecules a) b)
journal, December 1982
- Reynolds, Peter J.; Ceperley, David M.; Alder, Berni J.
- The Journal of Chemical Physics, Vol. 77, Issue 11
Calculating rovibrationally excited states of H2D+ and HD by combination of fixed node and multi-state rotational diffusion Monte Carlo
journal, February 2016
- Ford, Jason E.; McCoy, Anne B.
- Chemical Physics Letters, Vol. 645
Comparison of quantum Monte Carlo with time-dependent and static density-functional theory calculations of diamondoid excitation energies and Stokes shifts
journal, December 2011
- Marsusi, F.; Sabbaghzadeh, J.; Drummond, N. D.
- Physical Review B, Vol. 84, Issue 24
Nodeless wave functions and spiky potentials
journal, May 1981
- Hunter, Geoffrey
- International Journal of Quantum Chemistry, Vol. 19, Issue 5
Density-density functionals and effective potentials in many-body electronic structure calculations
journal, June 2008
- Reboredo, F. A.; Kent, P. R. C.
- Physical Review B, Vol. 77, Issue 24
Excited states of incipient Wigner molecules
journal, May 2011
- Blundell, S. A.; Chacko, S.
- Physical Review B, Vol. 83, Issue 19
Quantum Monte Carlo Calculations of Nanostructure Optical Gaps: Application to Silicon Quantum Dots
journal, October 2002
- Williamson, Andrew J.; Grossman, Jeffrey C.; Hood, Randolph Q.
- Physical Review Letters, Vol. 89, Issue 19
Ground and excited electronic states of azobenzene: A quantum Monte Carlo study
journal, December 2010
- Dubecký, M.; Derian, R.; Mitas, L.
- The Journal of Chemical Physics, Vol. 133, Issue 24
Diffusion quantum Monte Carlo calculation of the quasiparticle effective mass of the two-dimensional homogeneous electron gas
journal, January 2013
- Drummond, N. D.; Needs, R. J.
- Physical Review B, Vol. 87, Issue 4
Using fixed-node diffusion Monte Carlo to investigate the effects of rotation-vibration coupling in highly fluxional asymmetric top molecules: Application to H 2 D +
journal, January 2013
- Petit, Andrew S.; Wellen, Bethany A.; McCoy, Anne B.
- The Journal of Chemical Physics, Vol. 138, Issue 3
Monte Carlo energy and variance-minimization techniques for optimizing many-body wave functions
journal, May 1999
- Kent, P. R. C.; Needs, R. J.; Rajagopal, G.
- Physical Review B, Vol. 59, Issue 19
A study of the fixed-node error in quantum Monte Carlo calculations of electronic transitions: The case of the singlet n→π∗ (CO) transition of the acrolein
journal, March 2009
- Bouabça, Thomas; Ben Amor, Nadia; Maynau, Daniel
- The Journal of Chemical Physics, Vol. 130, Issue 11
Spin contamination in quantum Monte Carlo wave functions
journal, June 1998
- Huang, Chien-Jung; Filippi, Claudia; Umrigar, C. J.
- The Journal of Chemical Physics, Vol. 108, Issue 21