THE JOURNAL OF CHEMICAL PHYSICS 135, 194105 (2011) Accurate potential energy surfaces with a DFT+U (R) approach Heather J. Kulik 1, a) and Nicola Marzari 2 1 Department of Chemistry, Stanford University, Stanford, California 94305, USA 2 Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom (Received 15 May 2011; accepted 24 October 2011; published online 18 November 2011) We introduce an improvement to the Hubbard U augmented density functional approach known as DFT+U that incorporates variations in the value of self-consistently calculated, linear-response U with changes in geometry. This approach overcomes the one major shortcoming of previous DFT+U studies, i.e., the use of an averaged Hubbard U when comparing energies for different points along a potential energy surface is no longer required. While DFT+U is quite successful at providing accurate descriptions of localized electrons (e.g., d or f) by correcting self-interaction errors of standard exchange correlation functionals, we show several diatomic molecule examples where this position-dependent DFT+U (R) provides a significant two- to four-fold improvement over DFT+U predictions, when compared to accurate correlated quantum chemistry and experimental references. DFT+U (R) reduces errors in binding energies, frequencies, and equilibrium bond lengths by apply- ing the linear-response, position-dependent U (R) at each configuration considered. This extension is most relevant where variations in U are large across the points being compared, as is the case with covalent diatomic molecules such as transition-metal oxides. We thus provide a tool for deciding whether a standard DFT+U approach is sufficient by determining the strength of the dependence of U on changes in coordinates. We also apply this approach to larger systems with greater degrees of freedom and demonstrate how DFT+U (R) may be applied automatically in relaxations, transition- state finding methods, and dynamics. © 2011 American Institute of Physics. [doi:10.1063/1.3660353] I. INTRODUCTION Transition metals play key roles in a variety of biologi- cal and inorganic complexes, but theoretical understanding of these systems is often limited by the shortcomings of first- principles computational approaches. Exchange-correlation functionals that incorporate approximations based on the Hubbard model, 1–4 referred to generally as DFT+U, have been widely used in the solid-state physics community to treat strongly correlated systems. 5, 6 We have demonstrated 7–10 that such an approach can very accurately treat transition- metal complexes, 11, 12 where only one or few transition-metals are involved (see Ref. 13 for a comparison to different functionals). DFT+U achieves high accuracy by correcting self- interaction errors of standard local or semi-local exchange correlation functionals with Hartree-Fock-like treatments on a localized set of atomic orbitals. In addition, the appropriate strength of this local correction can be determined fully from a first-principles, linear-response formulation. 4 A rotationally invariant formulation of DFT+U (Refs. 3, 4, and 7) adds a Hubbard term of the form E U = U 2 I,σ Tr [n Iσ (1 - n Iσ )], (1) where n Iσ is the occupation matrix of the localized man- ifold(s) at site I with spin σ . This functional form, which is tied to the exact correction needed for simple exchange- correlation functionals in the limit of an atomic system, 4, 14 a) Electronic mail: [email protected]. penalizes fractional occupations and approaches zero as n ap- proaches 0 or 1. We determine the occupations of localized atomic levels that enter into E U by projection onto an atomic basis. It has been shown 4 that U may be calculated directly from linear response, χ IJ = d 2 E dα I dα J = dn I dα J , (2) where χ IJ is the response function obtained from applying an arbitrary shift α J to the potential on the site J that results in a reorganization of the occupations n I on site I. 15 A similar ex- pression may be obtained for the non-interacting case, χ 0 , as a linear shift in the potential can still result in rehybridization that must be removed from our overall expression to deter- mine U. The final U value is then obtained as U = χ -1 0 - χ -1 , (3) where χ is simply a scalar if we are only interested in a single manifold and site or it becomes a matrix in the case of mul- tiple sites or manifolds. We stress that U is calculated fully from first-principles as a system-dependent property and not used as a fitting parameter in any way (as an example, see Table I). A recent extension to the DFT+U approach focuses on inclusion of inter-site interactions, 16 which may play an important role in cases where strong 3d hybridization occurs with neighboring atoms. 17 It is possible to calculate U exactly for any configura- tion, and we often find the variations in the linear-response U to be quite significant. In the worked example we de- scribe throughout, a 4 FeO + , internuclear separations over a 0021-9606/2011/135(19)/194105/10/$30.00 © 2011 American Institute of Physics 135, 194105-1 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp This article has been downloaded from and is intended for personal use only http://www.kuliklab.org
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THE JOURNAL OF CHEMICAL PHYSICS 135, 194105 (2011)
Accurate potential energy surfaces with a DFT+U(R) approachHeather J. Kulik1,a) and Nicola Marzari21Department of Chemistry, Stanford University, Stanford, California 94305, USA2Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
(Received 15 May 2011; accepted 24 October 2011; published online 18 November 2011)
We introduce an improvement to the Hubbard U augmented density functional approach known asDFT+U that incorporates variations in the value of self-consistently calculated, linear-response Uwith changes in geometry. This approach overcomes the one major shortcoming of previous DFT+Ustudies, i.e., the use of an averaged Hubbard U when comparing energies for different points alonga potential energy surface is no longer required. While DFT+U is quite successful at providingaccurate descriptions of localized electrons (e.g., d or f) by correcting self-interaction errors ofstandard exchange correlation functionals, we show several diatomic molecule examples where thisposition-dependent DFT+U (R) provides a significant two- to four-fold improvement over DFT+Upredictions, when compared to accurate correlated quantum chemistry and experimental references.DFT+U (R) reduces errors in binding energies, frequencies, and equilibrium bond lengths by apply-ing the linear-response, position-dependent U (R) at each configuration considered. This extension ismost relevant where variations in U are large across the points being compared, as is the case withcovalent diatomic molecules such as transition-metal oxides. We thus provide a tool for decidingwhether a standard DFT+U approach is sufficient by determining the strength of the dependence ofU on changes in coordinates. We also apply this approach to larger systems with greater degrees offreedom and demonstrate how DFT+U (R) may be applied automatically in relaxations, transition-state finding methods, and dynamics. © 2011 American Institute of Physics. [doi:10.1063/1.3660353]
I. INTRODUCTION
Transition metals play key roles in a variety of biologi-cal and inorganic complexes, but theoretical understanding ofthese systems is often limited by the shortcomings of first-principles computational approaches. Exchange-correlationfunctionals that incorporate approximations based on theHubbard model,1–4 referred to generally as DFT+U, havebeen widely used in the solid-state physics community to treatstrongly correlated systems.5, 6 We have demonstrated7–10
that such an approach can very accurately treat transition-metal complexes,11, 12 where only one or few transition-metalsare involved (see Ref. 13 for a comparison to differentfunctionals).
DFT+U achieves high accuracy by correcting self-interaction errors of standard local or semi-local exchangecorrelation functionals with Hartree-Fock-like treatments ona localized set of atomic orbitals. In addition, the appropriatestrength of this local correction can be determined fully froma first-principles, linear-response formulation.4 A rotationallyinvariant formulation of DFT+U (Refs. 3, 4, and 7) adds aHubbard term of the form
EU = U
2
!
I,!
T r[nI! (1 ! nI! )], (1)
where nI! is the occupation matrix of the localized man-ifold(s) at site I with spin ! . This functional form, whichis tied to the exact correction needed for simple exchange-correlation functionals in the limit of an atomic system,4, 14
a)Electronic mail: [email protected].
penalizes fractional occupations and approaches zero as n ap-proaches 0 or 1. We determine the occupations of localizedatomic levels that enter into EU by projection onto an atomicbasis.
It has been shown4 that U may be calculated directly fromlinear response,
"IJ = d2E
d#I d#J
= dnI
d#J
, (2)
where " IJ is the response function obtained from applying anarbitrary shift #J to the potential on the site J that results in areorganization of the occupations nI on site I.15 A similar ex-pression may be obtained for the non-interacting case, "0, asa linear shift in the potential can still result in rehybridizationthat must be removed from our overall expression to deter-mine U. The final U value is then obtained as
U = "!10 ! "!1, (3)
where " is simply a scalar if we are only interested in a singlemanifold and site or it becomes a matrix in the case of mul-tiple sites or manifolds. We stress that U is calculated fullyfrom first-principles as a system-dependent property and notused as a fitting parameter in any way (as an example, seeTable I). A recent extension to the DFT+U approach focuseson inclusion of inter-site interactions,16 which may play animportant role in cases where strong 3d hybridization occurswith neighboring atoms.17
It is possible to calculate U exactly for any configura-tion, and we often find the variations in the linear-responseU to be quite significant. In the worked example we de-scribe throughout, a 4$ FeO+, internuclear separations over a
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194105-2 H. J. Kulik and N. Marzari J. Chem. Phys. 135, 194105 (2011)
TABLE I. Values of linear-response U0 (in eV) for the lowest quartet stateof diatomic molecule FeO+ calculated at several internuclear separations,rFe-O (in Å).
rFe-O (Å) 1.58 1.75 1.93 2.12 2.29 2.43U0 (eV) 6.25 5.27 4.26 3.25 2.22 1.24
1 Å range have values of U that differ by over 5 eV (seeTable I).18 Nevertheless, a drawback of the DFT+U ap-proach retains significance: there is no constant-U global po-tential energy surface, but there are instead locally correctbut distinct potential energy surfaces (PES), as we show inFigure 1. One curve, e.g., U = 6 eV (shown in dark blue), islocally correct for points where the bond distance is slightlyless than the minimum, but it is incorrect elsewhere, e.g., atdissociation. In order to obtain a highly accurate global poten-tial energy surface, we need to take into account the fact thatU is dependent on R, but previous approaches have merelyincorporated these variations by obtaining global or local av-erages of U in a constant-U DFT+U approach.
It is also possible to iteratively calculate U to obtain astructurally consistent U: the generalized gradient approxima-tion (GGA) optimized structure is typically used to determinethe linear-response U0, but we can then relax the structure atthat U0 and recalculate a linear-response U on this new struc-ture to incorporate structural variations into an improved cal-culation of U.19 The practical complications of a constant, av-eraged DFT+U approach include underestimation of reactionbarriers and dissociation energies and stabilization of weaklybound geometries. Inclusion of a structurally consistent U canimprove upon the latter complication but it cannot address theother two issues. We now introduce an improved approachthat addresses the concerns of a constant-U DFT+U approachby incorporating variations of U with position, R, that we re-fer to as DFT+U (R). In DFT+U (R), we interpolate acrossfirst derivatives of the DFT+U energy at different values of U(see Figure 2). We also require the calculation of the variationof U with R either directly at each position being comparedor through calculation of dU/dR, as we will later show ispossible.
One can write the correct expression for the total firstderivative of the energy in a manner that explicitly incorpo-rates variations of U with R,
dE
dR= %E
%R+ %E
%U
dU
dR, (4)
where the first term is simply the Hellmann-Feynman forcecalculated in any standard electronic structure calculation20
and the second term is the difference in DFT+U (R). Thederivative of the U-dependent component of the total energyexpression with respect to U is simply,
dE
dU= %E
%U= 1
2Tr[n(1 ! n)], (5)
because the second term in the standard definition of the to-tal derivative dE/dU is %E/%&%&/%U, which is zero fromHellmann-Feynman. Using the correct expression of dE/dR,
FIG. 1. Total energies of 4FeO+ (left) for different values of U in theDFT+U approach as shown by results with U varying from 0 to 6 eV (colorkey shown in inset).
we integrate to get an expression for 'E,
'E ="
R
dE
dRdR =
"
R
#%E
%R+ %E
%U
dU
dR
$dR. (6)
In Figure 2, the first derivatives of constant-U DFT+Utotal energies for a 4$ FeO+ all follow a similar trend. Thecorrect expression for the DFT+U (R) first derivative incor-porates variations of U with R and is a smooth interpolationbetween constant-U curves in Figure 2. Upon integration, weobtain a smooth potential energy curve that incorporates vari-ations in U and, in this case, resembles a high U DFT+Ucurve at short internuclear separation and a low U curve athigh internuclear separation as a result of the form of the vari-ation of U with position.
In this article, we will demonstrate that we can not onlycalculate changes in U with interatomic distance but also de-rive a gradient of U that enables a calculation of the po-tential need for a DFT+U (R) approach for any given sys-tem. We will apply this approach to several small diatomicmolecules of varying composition as well as demonstratethe extension to polyatomic systems. This U (R) extension tostandard DFT+U is also related to ideas of range-separationin exchange-correlation functionals that vary the amount ofHartree-Fock exchange in the functional based on a distanceconstraint.21, 22
II. METHODS
Plane-wave, density functional calculations were com-pleted with the QUANTUM-ESPRESSO package23 using thePerdew-Burke-Ernzerhof (PBE) (Ref. 24) GGA. We aug-mented this standard GGA functional with a self-consistent,linear-response, Hubbard U term, as previously outlined.4, 7
New extensions to the standard DFT+U approach presentedhere are also implemented in this code. Ultrasoft pseudopo-tentials are used with a plane-wave cutoff of 30 Ry for thewavefunction and 300 Ry for the charge density to ensureforces and spin state splittings are converged with respect tobasis set size.25 Semi-core states (3s and 3p) are included in
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194105-3 Accurate PES with a DFT+U(R) approach J. Chem. Phys. 135, 194105 (2011)
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4Fe-O Distance (Å)
-10
-8
-6
-4
-2
0
2
4
dE/d
R (e
V/Å
)
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4Fe-O Distance (Å)
0 U 6
0.0
1.0
2.0
3.0
4.0
Rel
ativ
e E
nerg
y (e
V)
FIG. 2. The derivatives of total energy, E, with respect to R for 4FeO+ at constant values of U are shown at left. An interpolated derivative is obtained fromincorporating variations in U (dashed black curve). The graph at right shows the integrated result of the U (R) curve along-side constant-U (0 eV in red and6.25 eV in blue) curves for comparison. The color key for all curves is shown in the bottom right.
the valence alongside the 3d and 4s states for the early- andmid-row transition-metals (Sc-Fe), while for late transition-metals (Co-Zn), only the 3d and 4s states are included in thevalence.26 The harmonic frequencies are obtained from a lin-ear fit to the first derivative of the energy at equilibrium.
Post-Hartree-Fock (HF) approaches are employed to pro-vide an accurate but computationally expensive referencefor the density-functional calculations. The single-referencemethod, coupled cluster with singles, doubles, and pertur-bative triples [CCSD(T)], is carried out as implemented inGAUSSIAN (Ref. 27) with a Pople-style 6-311++G(3df,3pd)basis set. Where literature correlated quantum chemistry (CC)results are available, those are used in place of our own calcu-lations and the level of theory is noted.
III. CALCULATING dU/dR
It is useful to both understand and predict the behaviorof the linear-response value of U with variations in position,R. In cases we have studied thus far,8, 10 dU/dR monotoni-cally decreases in an approximately linear fashion for valuesof bond distance, r, stretched beyond the equilibrium value,re. This behavior is likely derived from the decay of hybridiza-tion in orbitals as the atoms in a molecular bond are separated.One could therefore calculate U at several points and interpo-late to get an approximate expression of the variation of Uwith position. However, we will show that it is possible toobtain dU/dR from quantities already calculated during thelinear-response U calculation at fixed position.
Recall that the Hubbard U is the unphysical curvature ofthe energy with respect to occupations and that the derivativeof U with respect to coordinates takes on the form,
dU
dR= d3E
dRdn2. (7)
Generally, we do not measure the curvature of the energywith occupations directly but we instead calculate the linear-response U by measuring the reorganization of occupationswith respect to a rigid potential shift on the levels of the lo-
calized manifold,
" = dn
d#; U = "!1
0 ! "!1. (8)
In order to determine the dependence of U on internuclearseparation, we can first look at how " , the response function,varies with R
d"
dR= d
dR
#dn
d#
$= d
d#
#dn
dR
$" d
d#
#%n
%R
$. (9)
Rearrangement of the derivatives enables us to identify aquantity we already calculate: the derivative of the localizedmanifold’s occupations with respect to internuclear separa-tion, %n/%R.28 This quantity is already calculated as a com-ponent in the Hubbard-dependent forces, but it is a partialderivative that we approximate in this case to be equivalentto the total derivative dn/dR, which would normally includean additional term comparable to (%n/%&)(%&/%R).28 Bychoosing to describe %n/%R in each iteration, we can thenexamine the dependence of this occupation derivative on thevalue of #.
We can rearrange the expression for the dependence of" on R to obtain instead an expression for dU/dR by firstrecalling the relationship between U and " ,
dU
dR= d
dR%"!1
0 ! "!1&. (10)
The relationship between d"/dR and d"!1/dR is straight-forward and this permits us to express dU/dR in terms ofquantities we already intend to calculate, " = dn/d# and%n/%R,
dU
dR= d
d#
#dn
dR
$ %"!1&2 ! d
d#
#dn0
dR
$ %"!1
0
&2. (11)
We may calculate dU/dR at the same time as we calculate thelinear-response U, using quantities that are already calculatedduring the course of a standard calculation to determine thevalue of the linear-response U.
We now use the a 4$ FeO+ molecule as an example inwhich we calculate dU/dR to predict U over a range of in-ternuclear separations. Here, we start by simply determining
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194105-4 H. J. Kulik and N. Marzari J. Chem. Phys. 135, 194105 (2011)
1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6Internuclear Separation (Å)
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45!
(eV
)!0!
FIG. 3. The calculated values of the bare (black circles) and self-consistent(red circles) response functions, " , compared to a forward difference approx-imation on the calculated value of d"/dR (from Eq. (9)) at 1.6 Å internuclearseparation (dashed lines) for a 4$ FeO+.
d"/dR and d"0/dR (see Figure 3). The approximation of "
as linear in R is a poor one as it cannot incorporate varia-tions in " at longer distances from the point at which d"/dRwas calculated. However, the necessary inversion that relates" with U produces a net cancellation in the errors betweenthe bare and self-consistent response functions that gives usexcellent determination of the position dependence of U overa 1 Å range (Figure 4). Net cancellation from the differencein the response functions likely also minimizes any errors in-troduced by taking the partial derivative of the occupations,%n/%R, as equivalent to the total derivative, dn/dR in ourderivation. Therefore, the formal approximation made in ourdetermination of dU/dR has little, if any, practical implica-tions on the quality of the prediction of the dependence of Uon R.
Ultimately, for highly accurate interpolation of the po-tential energy surface, one may wish to calculate both U anddU/dR directly for several points along a relevant coordi-nate. These quantities, alongside an additional metric we will
1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6Internuclear Separation (Å)
0
1
2
3
4
5
6
Hub
bard
U (e
V)
Ufwd.diff.U0
FIG. 4. Predicted U (blue dashed line) from forward difference approxima-tion with dU/dR calculated (from Eq. (11)) at 1.6 Å Fe–O distance is com-pared against actual U0 values (black circles), for the a 4$ state of FeO+ overa range of internuclear separations.
TABLE II. The linear-response U0 (eV), self-consistent Uscf (eV), dU/dR(eV/Å), distance from equilibrium at which U0 is half of U0|re , 'rU1/2 (Å),compared for several molecules.
Molecule U0 Uscf dU/dR 'rU1/2
2(+ CoC 5.0 4.8 !4.0 0.62(! CrN 4.3 4.3 !2.3 0.92' FeN 5.7 5.9 !5.8 0.63' TiO 5.7 6.0 !3.1 0.94(! VO 4.4 4.5 !2.9 0.84$ FeO+ 6.3 5.5 !5.0 0.63' ScF 2.5 2.6 !2.5 0.56(+ CrF 1.8 2.0 !0.1 9.05(+ MnF 2.4 2.2 !4.8 0.24' FeF 2.3 2.6 !3.4 0.36' FeF 1.5 1.8 !3.2 0.2
introduce shortly, also serves as a measure of how importanta DFT+U (R) approach might be for a given system.
IV. RESULTS OF DFT+U(R) ON MOLECULES
We study the full potential energy curves of ten repre-sentative diatomic molecules that span early to late transition-metal carbides, nitrides, oxides, and fluorides of varying spinand total orbital angular momentum (Table II).29 In order toassess the relative importance of variations in linear-responseU with change in internuclear separation for these molecules,we determine a few quantities that help us to measure the rel-evance of a variable-U approach.
We first calculate the linear-response U0 and self-consistent Uscf
29 at the equilibrium bond length as determinedby GGA; these quantities are used for all calculations in thecurrent implementation of DFT+U (Table II). In addition, wecalculate dU/dR, the variation of U with position as calcu-lated at the GGA equilibrium bond length. While the car-bides, nitrides, and oxides generally have higher values ofUscf than the fluorides, the values of dU/dR are comparable(3–5 eV/Å). The relative variation of U with position may bebest measured by a new quantity,
'rU 12
= r|U= 12 U0
! r0,GGA = !U0
2 dUdR
, (12)
where 'rU1/2 is the displacement from equilibrium bondlength (in Å) at which the new value of linear-response U ishalf of the original Uscf value. If the value of 'rU1/2 is small,the DFT+U (R) approach is likely to be important. We pro-vide below some benchmark values of this quantity for severaldifferent diatomic molecules.
The smallest values of 'rU1/2 are obtained for a sub-set of the fluorides at around 0.2–0.3 Å, while more cova-lent molecules have 'rU1/2 in the range of 0.6-0.9 Å (seeTable II). A reduction of U to half of its original valueis a substantial change, and 'rU1/2 under 1.0 Å indicatesDFT+U (R) will provide significantly different results fromstandard DFT+U. The outlier in our data set is 6(+ CrF: thevalue of U changes very little from equilibrium to dissoci-ation, and a standard DFT+U treatment must provide nearly
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194105-5 Accurate PES with a DFT+U(R) approach J. Chem. Phys. 135, 194105 (2011)
identical results to DFT+U (R). This flat dU/dR for 6(+ CrFis likely due to the fact that U is of a similar value for theequilibrium and nearly dissociated products, despite signifi-cant differences in occupations at these respective points. Theeasily calculated or predicted metrics, 'rU1/2 and dU/dR, areindicators of the potential utility of DFT+U (R).
Of the diatomic molecules we have previously stud-ied with standard DFT+U, the transition-metal carbides andtransition-metal nitrides have had the poorest agreementbetween simulation and experiment where energetics andstructural properties were available.10 Three representativemolecules, X 2(+ CoC, a 2(! CrN, and X 2' FeN havemoderate values of U0 and Uscf in a relatively narrow range of4.3–5.9 eV (see Table II). The dU/dR metric is larger forCoC and FeN (!4.0 and !5.8 eV/Å, respectively) than forCrN (!2.3 eV/Å), and the corresponding 'rU1/2 metric isequivalent for the isoelectronic CoC and FeN at 0.6 eV/Å,while CrN is a slightly longer 0.9 eV/Å.
Structural properties of the X 2' FeN ground statechange dramatically from their GGA values upon inclusionof a Uscf of 5.5 eV (Table III). The bond distance elongatesby 0.24 Å, the harmonic frequency is reduced to half of theGGA value, and the dissociation energy is reduced by 1.2 eV.These structural changes are due to changes in the molecu-lar orbitals, which have the nominal electron configuration! 2)4*3! (4s)2. The population of a minority spin ! (4s) bond-ing orbital declines with increasing U and the population sub-sequently increases in previously unoccupied majority spin!* and )* orbitals. This net increase in spin on iron corre-sponds to a decrease in covalent character that is responsiblefor the longer bond in DFT+Uscf, av calculations. With respectto an average-U approach, DFT+U (R) significantly improvesthe structure (re = 1.64 Å, +e = 674 cm!1, De= 2.9 eV) whencompared against CCSD(T) (re = 1.57 Å, +e = 624 cm!1, De= 2.9 eV) and experiment (re = 1.59 Å).
Cobalt monocarbide is isoelectronic with FeN but hasan X 2(+ ground state with a valence electron configura-tion of ! 2)4*4! (4s)1. Bond elongation over GGA is a mod-erate 0.02 Å for both DFT+Uscf, av and DFT+U (R) andchanges in +e and De are comparable between the methods(Table III). The absence of a high-energy, minority spin ! (4s)orbital in this symmetry explains the lack of large scale struc-
TABLE III. Structural properties of M(C/N) calculated at the GGA,DFT+Uscf, av, DFT+U (R), and correlated quantum chemistry (CC) levelcompared against experiment. Experimental properties for CoC (Refs. 30 and31) and FeN (Ref. 32) and MRCI results for CoC (Ref. 33) are from literature.
GGA +Uscf +U (R) CC Expt.
CoC re 1.51 1.53 1.53 1.55 1.56X 2(+ +e 1092 981 990 965 934
De 4.2 3.5 3.6 3.5 ...
CrN re 1.53 1.56 1.56 1.56 ...a 2(! +e 1110 892 1018 1090 ...
De 4.8 2.7 2.8 2.2 ...
FeN re 1.55 1.79 1.64 1.57 1.59X 2' +e 960 438 674 624 ...
De 3.5 2.3 2.9 2.9 ...
tural changes with increasing U in CoC as compared againstdramatic changes in X 2' FeN. The DFT+Uscf, av approachyields good agreement with correlated quantum chemistry33
and experiment,30, 31 and the more rigorous DFT+U (R) pre-serves these features.
The a 2(! state of CrN has a longer displacement met-ric than CoC or FeN but is still relatively short at 0.9 Å.The DFT+Uscf, av results decrease the harmonic frequency byover 200 cm!1, reduce the dissociation energy by 2 eV, whileelongating the bond length by only 0.03 Å with respect toGGA (Table III). Such a variation in the harmonic frequencyis likely derived from the ! 2)4*2! (4s)1 electron configura-tion, where ! (4s) is minority spin, in a manner similar to thatfor X 2' FeN. These results on CrN highlight the fact thatthe displacement metric reflects only averaged distance de-pendence and does not indicate the character of the orbitals(bonding, non-bonding, or anti-bonding) that change occupa-tion most significantly over changes in coordinates. Use of aDFT+U (R) approach compensates for some of the excessivedecrease in +e by standard DFT+Uscf, av in improved agree-ment with correlated quantum chemistry while also providingconsistent values of re and De (Table III). The significant im-provement in structural properties for the carbides and nitridesdiscussed here suggests that a DFT+U (R) approach is criticalfor this class of compounds.
Transition-metal oxides are relatively well studied, andwe have previously studied a large number of these moleculeswith the constant-U DFT+U approach.7, 8, 10 We now considera few representative transition-metal oxides: X 3' TiO, X 4(!
VO, and a 4$ FeO+. Of the three molecules, FeO+ has theshortest displacement metric of 0.6 Å and the largest valueof U0 (6.3 eV). However, the range of values of U (around4.5 to 6.5) and 'rU1/2 (0.6 to 0.9 Å) is narrow, and we ex-pect all three molecules to behave differently when comparingDFT+U (R) and DFT+Uscf, av approaches.
We have previously studied in depth the a 4$ state ofFeO+,7, 8 which has a bond length of 1.58 Å and harmonicfrequency of 1038 cm!1 when calculated with GGA. Inclu-sion of a U0= 6.3 eV (as calculated at the GGA re) decreasesthe apparent bond order of this system resulting in a 0.22 Åbond elongation and lowering the harmonic frequency by over400 cm!1 (Table IV). The DFT+U (R) result, strongly moti-vated here by a large dU/dR, yields much improved struc-tural properties (re = 1.75 Å, +e = 720 cm!1) versus coupledcluster values (re = 1.70 Å, +e= 705 cm!1). Additionally,the dissociation energy which is significantly underestimatedfor DFT+Uscf, av at 1.7 eV with respect to the CCSD(T) valueof 3.0 eV, is in much better agreement with DFT+U (R) (De= 3.2 eV). As an indication of the smooth quality of the po-tential energy curve obtained from DFT+U (R) and its rela-tionship to the GGA and standard DFT+U potential energycurves, we plot all three in the right panel of Figure 2. TheU (R) curve at small r strongly resembles the DFT+U curve,while it more closely tracks with GGA for larger r.
The ground states of TiO and VO, 3' and 4(!, re-spectively, are approximately described by M2 +-O2 !-derivedconfigurations, with ! 1*x occupation (x = 1 for Ti and 2 forV). Applying DFT+Uscf, av on either state penalizes ! occupa-tion in favor of non-bonding and anti-bonding states. In turn,
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194105-6 H. J. Kulik and N. Marzari J. Chem. Phys. 135, 194105 (2011)
TABLE IV. Structural properties from GGA, DFT+Uscf, av, andDFT+U (R) of several transition-metal oxides compared against correlatedquantum chemistry (CC) and experimental results. Experimental results forTiO (Refs. 34 and 35) and VO (Ref. 36) and MRCI results for VO (Ref. 37)were obtained from the literature.
GGA +Uscf +U (R) CC Expt.
TiO re 1.62 1.65 1.65 1.63 1.62X 3' +e 1040 961 977 1014 1009
De 7.4 6.9 7.0 6.9 7.0
VO re 1.60 1.63 1.62 1.58 1.59X 4(! +e 1053 951 985 1010 1011
De 6.8 6.5 6.6 6.5 6.5
FeO+ re 1.58 1.80 1.75 1.70 ...a 4$ +e 1038 621 720 705 ...
De 4.1 1.7 3.2 3.0 ...
the DFT+Uscf, av structures exhibit bond elongation by about0.03 Å and decreases in both harmonic frequency (about80–100 cm!1) and dissociation energy (0.3-0.5 eV), asshown in Table IV. The results from DFT+U (R) for bothstates, which also have nearly identical values of dU/dR= !3 eV/Å, improve upon the DFT+Uscf, av harmonic fre-quency by around 30 cm!1. Overall, most quantities are inimproved agreement with both quantum chemistry and ex-periment when calculated with DFT+U (R) over the stan-dard DFT+Uscf, av approach (Table IV). Despite small val-ues of linear-response U associated with strong ionic char-acter in transition-metal fluorides, these molecules are quitesensitive10 to the value of U, as very small 'rU1/2 metrics of0.2-0.3 Å suggest. We present results here of our investigationon several representative transition-metal fluorides: a 3' ScF,X 6(+ CrF, a 5(+ MnF, and both the a 4' and X 6' FeF.
Structural properties of a 3' ScF obtained with GGAare in good agreement with both CCSD(T) and experiment(Table V). Although GGA results demonstrate typicaloverbinding, the equilibrium bond distance, harmonic fre-quency, and dissociation energy are within 0.03 Å, 21 cm!1,and 0.1 eV of the experimental values. The DFT+Uscf, av re-sults (Uscf = 4.0 eV) are instead underbinding, most notablyby underestimating the harmonic frequency by 46 cm!1 andthe dissociation energy by 0.8 eV. The electron configura-tion of a 3' ScF is nominally *1! 1, where the ! orbital is astrong mixture of 4s and 3dz2 states. The most significant dif-ference in the population of 3d-derived states after inclusionof a Hubbard term is a decline in the overall covalent char-acter of these orbitals. The DFT+U (R) results recover someof the good structural properties of GGA, while providingeven better agreement with experiment: a bond length within0.01 Å, harmonic frequency within 5 cm!1, and a consistentdissociation energy.
The midrow transition-metal fluorides, X 6(+ CrF anda 5(+ MnF, differ only by an additional minority spin ! or-bital in MnF. Both states have the approximate electron con-figuration ! x)2*2 (x= 1 for Cr, 2 for Mn). For both cases,DFT+Uscf, av calculations with a U around 2 eV produce0.03–0.04 Å longer bonds, 30-40 cm!1 lower harmonic fre-quencies, and 0.3-0.4 eV lower dissociation energies than
TABLE V. Structural properties from GGA, DFT+Uscf, av, and DFT+U (R)of several MF compared against correlated quantum chemistry (CC) and ex-periment. Experimental values for ScF, (Refs. 38 and 39) CrF, (Ref. 40) MnF,(Ref. 41) and FeF, (Ref. 42) and MRCI values for ScF (Ref. 43) and CrF,MnF, and FeF (Refs. 44 and 45) are from literature.a
GGA +Uscf +U (R) CC Expt.
ScF re 1.84 1.90 1.88 1.87 1.87a 3' +e 670 603 644 588 649
De 5.6 4.7 5.5 5.7 5.5
CrF re 1.79 1.82 1.81 1.80 1.78X 6(+ +e 667 631 641 655 664
De 4.8 4.4 4.6 4.7 4.6
MnF re 1.77 1.81 1.80 1.81 1.79a 5(+ +e 653 626 642 639 646
De 3.9 3.6 3.8 4.0 ...
FeF re 1.74 1.76 1.75 1.78 1.74a 4' +e 657 642 669 663 684
De 4.5 4.4 4.4 4.1 ...
FeF re 1.78 1.80 1.80 1.80 1.78X 6' +e 635 625 634 650 663
De 4.4 4.4 4.4 4.7 4.6
aGround state experimental values were obtained from Ref. 31 when not otherwisespecified.
those calculated with GGA (Table V), and these structuralchanges are much larger than those undergone by oxidesover a similar range of values of U.8 Inclusion of U-variationthrough DFT+U (R) recovers improved agreement with cor-related quantum chemistry and experiment (see Table V).
The ground state, X 6', and lowest lying excited quar-tet state, a 4', of FeF both have the electron configuration! 2)2*3, with the spin of the ! orbitals either being parallelor anti-parallel, respectively. The geometric structure of thesetwo states is similar when calculated with GGA (Table V).The DFT+U (R) structural properties are in good agreementwith both experiment and correlated quantum chemistry, no-tably by increasing the harmonic frequency of a 4' over thatobtained from GGA alone and in improved agreement withexperiment.
Errors for the transition-metal carbides, nitrides, oxides,and fluorides GGA and DFT+U with both constant Uscf andvariable-U (R) approaches are reported in Table VI. The ab-solute average and maximum errors are all reduced by two- tofour-fold with DFT+U (R) from their values in DFT+Uscf, av.The harmonic frequencies and dissociation energies from theDFT+U (R) approach are superior to the other two methodsand the bond lengths are in closer agreement with GGA andexperiment. We note that a variable-U will always produce amore rigorously correct result over the standard averaged Uapproach, but bond lengths may be further improved throughinclusion of inter-site, or “+V”, terms in the model Hubbard-like functional.16
A. State splittings from DFT+U(R) curves
The interplay between multiple states of differing spinor electronic character is vital in some cases, such as thosewhere reactivity is determined by surface crossing points. If
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194105-7 Accurate PES with a DFT+U(R) approach J. Chem. Phys. 135, 194105 (2011)
TABLE VI. Average and maximum errors for GGA, DFT+Uscf, av, andDFT+U (R) properties as referenced against experimental results. Bold in-dicates smallest error.
Average Absolute errors Maximum errors
GGA +Uscf +U (R) GGA +Uscf +U (R)
re 0.014 0.049 0.026 0.040 0.220 0.070+e 36 38 20 127 60 32De 0.24 0.26 0.06 0.40 0.80 0.20
the value of Hubbard U for both states remains very similar,it is straightforward to align the two states at a point in theregion of interest where the values of U are the same. In manycases, the value of U for the two states will differ over the re-gion of interest, but there are alternative alignment schemesthat are equally valid and based on the chemistry of the rele-vant systems.
In the case of diatomic molecules, if the two statesshare the same dissociation limit, alignment of the curves asr # $ is quite simple. An extension to larger molecules isalso possible if the important U (R) coordinates correspond todissociation events (e.g., molecular oxygen on different statesof porphyrins). If the dissociation limits differ, the two statescould instead be equivalent at the united atom limit, and analignment as r # 0 may instead be possible.46 An exampleof the united atom alignment applied to the DFT+U (R) 6'
and 4' FeF energy curves is given in Figure 5. Alignmentof the DFT+U (R) curves recovers a 6' # 4' splitting of0.60 eV, in remarkable agreement with the experimental valueof 0.62 eV (versus a standard DFT+Uscf, av approach whichpredicted a splitting of 0.39 eV). Ultimately, our aim is toalign the two DFT+U (R) curves in a manner that leads to thecancellation of the energetic error associated with the arbi-trary, U-dependent shifts in energy in DFT+U (R). Total en-ergies may be compared across two states at the same valueof U, and the reference point should be chosen to be one inwhich both states are in a similar bonding regime.
1.6 1.8 2Fe-F distance (Å)
0
0.5
1
1.5
Rel
ativ
e E
nerg
y (e
V)
1.6 1.8 2Fe-F distance (Å)
1.6 1.8 2Fe-F distance (Å)
GGA +Uscf +U(R)
FIG. 5. Relative energies of the X 6' (blue) and a 4' FeF (red) electronicstates for GGA (left), DFT+Uscf, av (middle), and DFT+U (R) (right). TheDFT+U (R) curves are aligned at the united atom limit.
B. Building variable-U potential energy surfacesfor polyatomic systems
Until this point, we have largely focused on the low-dimensional cases of diatomic molecules where R refers sim-ply to the interatomic distance between the two atoms thatcomprise the molecule. However, the DFT+U (R) approachmay be straightforwardly applied to a greater number of di-mensions in the reaction coordinate. In polyatomic systems,many of the potential degrees of freedom do not alter the oc-cupation matrix of a relevant transition-metal site and can thusbe ignored by DFT+U (R). In particular, we note that the di-rectly coordinated species are likely to have the most imme-diate effect on the value of U that we calculate, and typicaltransition metals are at most six-coordinate. In addition to thisrestriction, most reaction coordinates focus on the making orbreaking of only a few bonds. As an example, we considerthe simple reaction of H2 on FeO+ that we have previouslydescribed in detail.7 This four atom system has six internaldegrees of freedom, but we show in Fig. 6 that only threedegrees of freedom define the reaction step of interest. In par-ticular, we focus on the second, elimination step of this reac-tion where one hydrogen is transferred from iron to oxygento form a leaving water molecule. In this step, the hydrogentransfer is fully described by only three coordinates: (1) iron-oxygen bond distance (RFe!O), (2) transferring hydrogen toiron-oxygen bond midpoint distance (rH!X), and (3) the angleformed by bonds 1 and 2 (, ). In fact, rH!X is fully dependenton , and so we may describe the reaction instead in terms ofonly two variables. Still, for those two variables, one may de-fine a surface of values of U that, as shown in Figure 6, variesto a greater degree over , than over RFe!O, as is chemicallyintuitive.
In turn, we may define a smoothed variable-U poten-tial energy surface that incorporates these linear-responseU variations. The result is the variable-U PES will resem-ble GGA+U results for small , and GGA results for large, . For clarity, we report only the minimum energy path, aone-dimensional representation of all relevant coordinates,
FIG. 6. Comparison of the minimum energy path (MEP) for the second stepof the reaction of FeO+ with H2. The U(R) solution is displayed in gray withcolor-coded points alongside iso-U curves and coupled-cluster results (blackdashes). Also shown are insets of the key internal coordinates (right) and thecalculated U in terms of these variables (left).
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194105-8 H. J. Kulik and N. Marzari J. Chem. Phys. 135, 194105 (2011)
that we obtain with this approach (Fig. 6). The variable-U results for this system are in improved agreement withCCSD(T) over either standard DFT or DFT+Uscf, av. In thissmall molecule case, we explicitly calculated all the points onthe potential energy surface, but it is straightforward to alsoincorporate variation in U into the determination of forces.The result is that the extension of DFT+U (R) to transition-state finding techniques, such as nudged elastic band, istrivial.
V. CONCLUSIONS
We have introduced a new DFT+U (R) approach thatovercomes the one major shortcoming of previous DFT+Ustudies: an average of Hubbard U values is no longer re-quired when comparing points along a potential energy sur-face that possesses strong variation in U. We have shownseveral diatomic molecule examples where this position-dependent DFT+U (R) approach provides a two- to four-fold improvement over already quite good DFT+U results.The DFT+U (R) method reduces errors in binding energies,frequencies, and equilibrium bond lengths by applying theprecise, linear-response U for each configuration considered.This approach is likely critical for describing reaction coor-dinates that are defined by dramatic changes in hybridiza-tion for the relevant transition-metal center. For periodic sys-tems, where equilibrium bond distances are most relevantand bond-breaking events are more rare, our approach be-comes more equivalent to a structurally consistent DFT+Uapproach.19 Nevertheless, variations of U with strain in a crys-tal could be straightforwardly incorporated in order to obtainimproved DFT+U (R) stress calculations and variable cellsimulations. Further improvement upon DFT+U (R) bondlengths may be achieved most straightforwardly through in-clusion of inter-site, or “+V”, terms in the model Hubbard-like functional as has been recently introduced16 and demon-strated on molecules.17 We have proposed two metrics thatreveal the relative importance of a DFT+U (R) approach fora given system: (1) the change in U with change in coordi-nates and (2) the bond displacement that yields a new linear-response U that is half of the one obtained at the equilibriumbond distance. Importantly, we predict dU/dR from a hand-ful of additional quantities that are already determined in thestandard linear-response procedure for a single configuration.This DFT+U (R) extension to the already successful DFT+Uapproach may be straightforwardly applied with scaling com-parable to standard semi-local exchange-correlation function-als, thus permitting unprecedented accuracy in simulationsof transition metal complexes several hundreds of atomsin size.
ACKNOWLEDGMENTS
Support from ARO-MURI DAAD-19-03-1-0169(H.J.K., N.M.) is gratefully acknowledged. H.J.K. acknowl-edges helpful conversations with M. Cococcioni and A. H.Steeves N.M. acknowledges helpful conversations with S. deGironcoli.
APPENDIX A: AN ALTERNATIVE CALCULATION OFdU/dR FROM FORCES
We present an alternate method for calculating dU/dRby considering the relationship between the position depen-dence of U and the Hellmann-Feynman forces at a given posi-tion. Recall that the U we calculate is a measure of the secondderivative of the energy with respect to occupations,
U = d2E
dn2. (A1)
The position-dependence of U may therefore be recast interms of an occupation-dependent force,
dU
dR= d3E
dn2dR= !d2F
dn2, (A2)
since the total derivatives are, by definition, symmetric.28 Wedo not wish to directly constrain occupations to obtain thissecond derivative but rather work within the linear-responseframework of the DFT+U approach to obtain an analogousexpression of d2F/dn2 in terms of quantities that depend onthe rigid potential shift, #, and on the linear response func-tion, " . The Hellmann-Feynman forces are a function of oc-cupations, n, position, R, and rigid potential-shift #, while theoccupations themselves are also dependent upon # and R.
Using the multivariable chain rule for dF/d# gives us
dFd#
= dFdn
dn
d#, (A3)
where the single expression on the right-hand side occurs asa result of both # and R being independent variables. Rear-rangement of this expression in terms of dF/dn permits us todifferentiate again,
d
dn
#dFdn
$= d
dn
#dFd#
d#
dn
$. (A4)
The final expression for d2F/dn2 in terms of only derivativesin # and expressions in " is
d2Fdn2
= d2Fd#2
#d#
dn
$2
+ dFd#
d2#
dn2. (A5)
-0.50 -0.25 -0.10 0 0.10 0.25 0.5"(eV)
0.234
0.236
0.238
0.240
F (R
y/a.
u.)
self-consistentbare
FIG. 7. The self-consistent (black circles) and bare (red squares) forces forquartet MnO2 in the linear structure with rMn!O = 1.95 Å versus the potentialshift, #. The points are fit with a quadratic regression shown in dashed lines.
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194105-9 Accurate PES with a DFT+U(R) approach J. Chem. Phys. 135, 194105 (2011)
1.6 1.8 2.0 2.2 2.4 2.6Internuclear Separation (Å)
0
1
2
3
4
5
6H
ubba
rd U
(eV
)
1.56 Å1.60 Å1.68 Å1.70 Å
FIG. 8. Calculated (black circles) compared to predicted (dashed lines) val-ues of U for the 4$ state of FeO+ over a number of values of Fe–O internu-clear separation (indicated in the legend).
A simplification to the above expression reveals that dU/dRmay be calculated from quantities that would be determinedin a standard linear-response U calculation,
dU
dR= d2F0
d#2
%"!1
0
&2 + dF0
d#
d"!10
dn! d2F
d#2("!1)2 ! dF
d#
d"!1
dn,
(A6)
where " is the linear response of the localized manifold andforces may be determined at varying # alongside the occu-pations. A shortcoming to this approach is that the secondderivatives of forces must be calculated, necessitating muchlarger values of # in order to capture curvature in the forces, atodds with the linear-response procedure for U determination.An example of the behavior of F with respect to # appears inFigure 7 along with a sample quadratic fit. The small val-ues of the derivatives on forces means that this procedureis more likely to suffer from numerical noise, which is thelikely source of scatter in dU/dR integrations in Figure 8.This calculation may be more robust for a number of caseswhere a curvature in the forces may be easily fit, but italso provides a point of comparison against the earlier de-scribed approach for determining dU/dR from derivatives inoccupations.
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