Constrained density functional theory (cDFT) is a versatile electronic
structure method that enables ground-state calculations to be performed subject
to physical constraints. It thereby broadens their applicability and utility.
Automated Lagrange multiplier optimisation is necessary for multiple
constraints to be applied efficiently in cDFT, for it to be used in tandem with
geometry optimization, or with molecular dynamics. In order to facilitate this,
we comprehensively develop the connection between cDFT energy derivatives and
response functions, providing a rigorous assessment of the uniqueness and
character of cDFT stationary points while accounting for electronic
interactions and screening. In particular, we provide a new, non-perturbative
proof that stable stationary points of linear density constraints occur only at
energy maxima with respect to their Lagrange multipliers. We show that multiple
solutions, hysteresis, and energy discontinuities may occur in cDFT.
Expressions are derived, in terms of convenient by-products of cDFT
optimization, for quantities such as the dielectric function and a condition
number quantifying ill-definition in multi-constraint cDFT.Comment: 15 pages, 6 figure

O'Regan, David D.

Teobaldi, Gilberto

English

arXiv

Constrained density functional theory (cDFT) is a versatile electronic structure method that enables ground-state calculations to be performed subject to physical constraints. It thereby broadens their applicability and utility. Automated Lagrange multiplier optimisation is necessary for multiple constraints to be applied efficiently in cDFT, for it to be used in tandem with geometry optimization, or with molecular dynamics. In order to facilitate this, we comprehensively develop the connection between cDFT energy derivatives and response functions, providing a rigorous assessment of the uniqueness and character of cDFT stationary points while accounting for electronic interactions and screening. In particular, we provide a new, non-perturbative proof that stable stationary points of linear density constraints occur only at energy maxima with respect to their Lagrange multipliers. We show that multiple solutions, hysteresis, and energy discontinuities may occur in cDFT. Expressions are derived, in terms of convenient by-products of cDFT optimization, for quantities such as the dielectric function and a condition number quantifying ill-definition in multi-constraint cDFT

O'Regan, David

DataCat: The Research Data Catalogue (University of Liverpool)

1 
Optimization of constrained density functional theory 
 
David D. O’Regan
1,*
 and Gilberto Teobaldi
2,3,† 
 
1
 School of Physics, CRANN and AMBER, Trinity College Dublin, Dublin 2, Ireland 
2 
Stephenson Institute for Renewable Energy, Department of Chemistry, University of 
Liverpool, Liverpool L69 7ZF, UK 
3
 Beijing Computational Science Research Center, Beijing 100094, China 
 
* daithi.o.riogain@tcd.ie 
†
 g.teobaldi@liv.ac.uk 
 
 
Abstract  
Constrained density functional theory (cDFT) is a versatile electronic structure method that enables ground-
state calculations to be performed subject to physical constraints. It thereby broadens their applicability 
and utility. Automated Lagrange multiplier optimisation is necessary for multiple constraints to be applied 
efficiently in cDFT, for it to be used in tandem with geometry optimization, or with molecular dynamics. In 
order to facilitate this, we comprehensively develop the connection between cDFT energy derivatives and 
response functions, providing a rigorous assessment of the uniqueness and character of cDFT stationary 
points while accounting for electronic interactions and screening. In particular, we provide a new, non-
perturbative proof that stable stationary points of linear density constraints occur only at energy maxima 
with respect to their Lagrange multipliers. We show that multiple solutions, hysteresis, and energy 
discontinuities may occur in cDFT. Expressions are derived, in terms of convenient by-products of cDFT 
optimization, for quantities such as the dielectric function and a condition number quantifying ill-definition 
in multi-constraint cDFT. 
 
 
2 
1. README for the data catalogue 
This archive contains open-data for Optimization of constrained density functional theory " by 
David D. O’Regan and Gilberto Teobaldi, accepted for publication in Physical Review B  
The open data is organized as follows:  
• Figure*.agr: xmgrace files and raw data of the figures in the paper 
• Figure*.txt: raw data of the figures in the paper 
Description of the Methods used to generate the data follows in Section 2. 
 
 
2. Methods 
The ONETEP linear-scaling simulation code
57
, was used to carry out cDFT calculations on the 
hydrogen molecule stretched to an internuclear distance of 3:2 a0, which, using the PBE 
functional
58
 with no spin-orbit coupling, lies just beyond the Coulson-Fischer point at which an 
open-shell singlet ground-state becomes favoured
59,60
 (see Fig. 1). A constraint was placed on the 
difference of spin magnetic moments, ΔM, between the two hydrogen atoms, defined on the basis 
of their isolated 1s valence pseudo-orbitals. The constraint target was set to ΔMc = 0 μB, and the 
Lagrange multiplier Vc was defined such that increasing its value increased the spin-dependent 
potential acting to decrease ΔM. It was found that the unpolarized, closed shell ”CS" state is meta-
stable, starting from which any non-zero value of Vc initiates a collapse to one of two symmetry-
related, degenerate open-shell ground-states, “OS(-)" and “OS(+)" defined in the caption of Fig. 1.  
 
In order to check the validity of our approach for analyzing functional interdependencies and 
derivatives, and to numerically illustrate our analytical findings, we performed a cDFT study on the 
nitrogen molecule shown in Fig. 2. This serves to illustrate a case in which, if the constraint energy 
Ec vanishes for a finite Lagrange multiplier Vc, the total-energy W achieves a maximum with 
respect to Vc. The total and constraint energies W and Ec exhibit different negative curvatures, and 
their the difference, the DFT component EDFT, necessarily exhibits a positive curvature around the 
ground-state minimum at Vc = 0 eV. For each value of the charge-constraining Lagrange multiplier, 
well converged BLYP
67,68
 ground-state energies and densities, with pseudized 1s states, were 
calculated using the ONETEP linear-scaling Kohn-Sham DFT code
57
. This code solves for the 
ground-state by optimising a minimal set of nonorthogonal generalized Wannier functions
69
 in 
situ. Each of these functions is expanded in an underlying variational plane-wave equivalent basis 
set and truncated within a prescribed cut-off sphere, in this particular case to a radius of 10 a0. 
This approach has been shown to offer finite-difference linear response properties with an 
accuracy matching that of conventional plane-wave DFT70. The constrained population was 
defined using the four 2s and 2p valence pseudo-orbitals of the isolated atom
71
. In the dimer, the 
resulting unconstrained ground state atomic occupancy was approximately 6:5 e, due to overlap 
between pseudo-orbitals. The charge of one of the nitrogen atoms was constrained, with the 
target occupancy set to Nc = 6:0 e. 
 
 
References 
57. C.-K. Skylaris, P. D. Haynes, A. A. Mostofi, and M. C. Payne, The Journal of Chemical Physics 122, 084119 
(2005).  
58. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).  
3 
59. C. Coulson and I. Fischer, Philos. Mag. 40, 386 (1949).  
60. A. Ruzsinszky, J. P. Perdew, and G. I. Csonka, The Journal of Physical Chemistry A 109, 11006 (2005). 
67. A. D. Becke, Phys. Rev. A 38, 3098 (1988).  
68. C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). 
69. C.-K. Skylaris, A. A. Mosto_, P. D. Haynes, O. Di_eguez, and M. C. Payne, Phys. Rev. B 66, 035119 (2002). 
71. The resulting constraint acts on the Kohn-Sham density matrix rather than on the density. Our analytical 
findindings extends to that case with minor notational changes.  


Optimization of constrained density functional theory

O'Regan, David D

University of Liverpool Repository

David D. O'Regan

Gilberto Teobaldi

Crossref

Constrained density functional theory (cDFT) is a versatile electronic
structure method that enables ground-state calculations to be performed subject
to physical constraints. It thereby broadens their applicability and utility.
Automated Lagrange multiplier optimisation is necessary for multiple
constraints to be applied efficiently in cDFT, for it to be used in tandem with
geometry optimization, or with molecular dynamics. In order to facilitate this,
we comprehensively develop the connection between cDFT energy derivatives and
response functions, providing a rigorous assessment of the uniqueness and
character of cDFT stationary points while accounting for electronic
interactions and screening. In particular, we provide a new, non-perturbative
proof that stable stationary points of linear density constraints occur only at
energy maxima with respect to their Lagrange multipliers. We show that multiple
solutions, hysteresis, and energy discontinuities may occur in cDFT.
Expressions are derived, in terms of convenient by-products of cDFT
optimization, for quantities such as the dielectric function and a condition
number quantifying ill-definition in multi-constraint cDFT

http://link.aps.org/pdf/10.1103/PhysRevB.94.035159

Optimization of constrained density functional theory

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