Thermal versus electronic broadening in the density of states of liquid water

Chemical Physics Letters 376 (2003) 68–74

www.elsevier.com/locate/cplett

Thermal versus electronic broadening in thedensity of states of liquid water

Patricia Hunt a, Michiel Sprik a,*, Rodolphe Vuilleumier b

a Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UKb Laboratoire de Physique Th�eeorique des Liquides, Universit�ee P. et M. Curie, 4 Place Jussieu, 75005 Paris, France

Received 18 April 2003; in final form 2 June 2003

Published online: 27 June 2003

Abstract

The one-electron density of states of liquid water computed from an ab initio molecular dynamics trajectory is

analyzed in terms of interactions between effective molecular orbitals localized on single molecules. These orbitals are

constructed from the occupied extended (Kohn–Sham) orbitals using the maximally localized Wannier function

method. Band positions are related to average orbital energies. The width of a band is resolved into contributions from

thermal fluctuations in the orbital energies and the electronic broadening due to intermolecular coupling. It is found

that the thermal and electronic broadening are of comparable magnitude with electronic broadening being the leading

effect.

� 2003 Elsevier Science B.V. All rights reserved.

1. Introduction

The Kohn–Sham (KS) approach to density

functional theory (DFT) was originally developed

for a selfconsistent treatment of the electronic

structure of crystalline solids. The one-electron

(KS) orbitals in these periodic systems are Bloch

states extended over all space. The corresponding

energy levels are grouped in bands, which in typ-ical semiconductor or ionic solids can be viewed as

superpositions of the valence states of the con-

stituent atoms. Combined with plane wave basis

* Corresponding author. Fax: +441223336362.

E-mail address: [email protected] (M. Sprik).

0009-2614/03/$ - see front matter � 2003 Elsevier Science B.V. All r

doi:10.1016/S0009-2614(03)00954-0

sets and (valence) pseudo potentials this approachhas evolved into a highly efficient tool for the

electronic structure computation of solids capable

of dealing with complex unit cells as for example

occur in oxides.

The same electronic structure machinery is ap-

plied in the ab initio molecular dynamics (MD)

method [1] to supercells of disordered condensed

molecular systems. A seemingly similar band pic-ture emerges. In the density of states of simple one-

component systems, such as pure liquid water

(Fig. 1), we can clearly recognize a broadened

version of the molecular orbital (MO) level dia-

gram of a single molecule. Similar to periodic

solids, the majority of one-electron states are

spread out over effectively all molecules in the cell.

ights reserved.

mail to: [email protected]

Fig. 1. Total density of occupied one-electron states for liquid

water as obtained by averaging over an ab initio molecular

dynamics trajectory of a system of 32 H2O molecules in a cubic

periodic box. The bands are labeled according to the corre-

sponding symmetry orbitals of a H2O molecule with C2v sym-

metry (the O1s core orbital has been omitted).

P. Hunt et al. / Chemical Physics Letters 376 (2003) 68–74 69

There are, however states, in particular near the

edges of the bands, which are localized on smaller

clusters of molecules. This suggests that perhaps

some of the models for electronic states in disor-dered solids are more appropriate for a description

of the electronic states in molecular liquids. A key

element in these models, such as the Anderson

model of substitutional disorder in alloys [2], is the

introduction of a statistical distribution of site

energies and (or) statistical variation in coupling

matrix elements. Whereas in alloys this disorder is

mostly quenched, in molecular liquids it is createdby thermal fluctuations.

In aqueous systems the thermal fluctuation of

interaction energies can be very large (in the range

of an eV). This raises the following question: to

what extent can the broadening of peaks in the

density of states of liquid water of Fig. 1 be at-

tributed to disorder in the molecular energy levels

and how much is due to band dispersion by elec-tronic interactions between molecules? The answer

to this question depends on what we define to be

the one-molecule states, because unlike the atomic

states in the tight binding hamiltonians used in

solid state physics for studying disorder (see, e.g.,

[3]), these states are not a priori available. The

approach we adopt here is to construct the one-

molecule states from the same set of delocalized

(canonical) KS states we are trying to analyze. The

basis of this method is the maximally localized

Wannier function scheme of Marzari and Van-

derbilt [4] and Silvestrelli et al. [5]. We exploit the

observation, validated by a growing series of ap-

plications, that for systems consisting of closedshell molecules the total number of Wannier

functions localized on or near the bonds or atoms

of a molecule is exactly equal to half the number of

valence electrons [6,7]. This unique set of Wannier

functions associated with a molecule can be used

as a basis for the effective molecular orbitals for

this molecule. A first application of this approach

can be found in [7].

2. Ab initio molecular dynamics method

The density of states of Fig. 1 has been com-

puted from an ab initio MD trajectory of pure

liquid water. This trajectory was generated using

the CPMD program [8]. The plane wave-pseudopotential implementation of the Kohn–Sham

method used in this code has been validated in

numerous previous studies of aqueous systems.

Similar to the majority of these simulations we

used the BLYP functional [9,10] as this combi-

nation of generalized gradients approximations to

exchange and correlation has been shown to give

good results for the structure and dynamics ofwater [6,11]. The specification of the various pa-

rameters of the simulation is standard and is es-

sentially identical to setting used in [11]. Again

the system consisted of 32 water molecules in a

simple cubic periodic cell of dimension 9.86 �AAcorresponding to the experimental density under

ambient conditions. The one-electron orbitals

were expanded in a plane wave basis up to anenergy cutoff of 70 Ry. For technical details of

the norm conserving pseudo potentials we refer to

[11]. The starting configuration was obtained

from a classical MD simulation using a model

force field. The ab initio trajectory was then

equilibrated to an average temperature of 300 K

using DFT Car–Parrinello for 2.9 ps. 20 struc-

tures were collected over the next 2.24 ps at equalintervals of 0.145 ps, hence the total length of the

trajectory was 5.140 ps.

70 P. Hunt et al. / Chemical Physics Letters 376 (2003) 68–74

All quantities discussed in this Letter were

computed from the electronic structure of these 20

configurations. The main property of interest is the

density of one-electron states which is formally

defined as

Dð�Þ ¼ hdð�� HHsÞi; ð1Þwhere HHs is the selfconsistent one-electron (KS)

hamiltonian. The spectrum of Fig. 1 was calcu-lated solving the secular equation HHswk ¼ �kwk for

each of the 20 sample configurations and binning

the resulting orbital energies �k in a histogram.

3. Effective single molecule orbitals

Maximally localized Wannier orbitals [4,5] canbe thought of as the periodic generalization of the

Boys localized orbitals [12] often employed by

quantum chemists in analyzing isolated molecules.

Thus the maximally localized Wannier orbitals

wnðrÞ are obtained from the canonical KS orbitals

wk by a unitary transformation U:

wn ¼Xk

Unkwk ð2Þ

such that the total spatial spread

X ¼Xn

hwnjr2jwnih

� hwn rj jwni2i

ð3Þ

is minimized. Summations in Eqs. 2 and 3 run over

the occupied subspace of KS orbitals. The same

convention is employed in all subsequent equa-

tions unless indicated otherwise. The crucial con-

tribution made by Marzari and Vanderbilt was

providing a definition of the expectation values for

position in Eq. (3) which is valid in periodic sys-

tems (crystals). In the present work we apply avariation of the original reciprocal space formal-

ism of [13,14]. This real space formulation [15] is

adapted to the single (C) point representation usedin the plane wave expansion of states in the large

supercells of disordered systems [5].

The KS orbital energies �k are uniquely definedas the diagonal elements of the KS hamiltonian HHs.

The corresponding canonical orbitals wk, however,are delocalized and the molecular picture is lost.

On the other hand, while the Wannier orbitals wn

are local, they are not eigenfunctions of HHs. This

suggests that an intermediate representation where

the orbitals are local to a molecule, but delocalized

within that molecule, is perhaps more suited for

the study of the chemistry of condensed molecular

systems. For the case of closed shell molecules,such a representation can easily be constructed

noting that each Wannier orbital can be unam-

biguously assigned to a specific molecule [6]. The

effective MOs are obtained by grouping together

all Wannier functions that belong to a single

molecule and then block-diagonalizing the Wan-

nier hamiltonian hnjHHsjmi within each molecularsubspace. This is the simple idea behind thescheme of [7]. The result is a new set of orbitals

which can be expressed as a linear combination of

Wannier orbitals:

/Ic ¼Xn

0TIc;nwn; ð4Þ

where the prime indicates that summation is re-

stricted to the Wannier functions constituting the

valence electrons of molecule I . The index c labelsthe different MOs of molecule I . Because of theblock diagonal form we can write for the I ¼ Jmatrix elements of the one-electron hamiltonian:

h/IcjHHsj/Ic0 i ¼ aIcdcc0 ; ð5Þ

where dcc0 is a Kronecker delta symbol stating that

off-diagonal elements in the same molecule sub-

space vanish. The diagonal matrix element aIc willbe interpreted as the energy of the effective energy

level c of molecule I in solution. The matrix ele-ments h/IcjHHsj/Jc0 i between two blocks I 6¼ J cor-respond to the electronic coupling between

molecules. Their magnitude is finite but small in

comparison to the intra molecular interaction that

has been eliminated by the partial diagonalization.

For the water system studied here, the localizedeffective MOs /Ic resemble the standard MOs for a

single C2v symmetry H2O molecule in vacuum.

Accordingly the orbitals will be labeled c ¼ 1A1;1B2; 2A1 and 1B1. Because the aqueous solution is

disordered these symmetry labels will not strictly

apply. However the shape and form of the orbitals

is very similar. Also the ordering in energy is

preserved. In practice, the orbitals of each mole-


cule are therefore assigned to one of these four

bands according to their relative orbital energies

aIc defined in Eq. (5).

4. Projected density of states

Localized molecular orbitals can be generated

in a number of ways. A popular method is the

natural bond orbital (NBO) analysis available in

quantum chemistry codes. This approach relies on

a projection on an optimal set of molecular orbi-

tals under minimal occupation of virtual orbitals

(for an application to intermolecular interactionssee [16]). Alternatively one could also use fragment

orbitals (both occupied and empty) which are

more directly obtained as the molecular orbitals in

vacuum of suitably defined subsets of atoms in the

condensed system (see for example [17]). After

orthogonalization, these local molecular orbitals

can then be used to represent the KS matrix. NBO

or fragment analysis is often combined with energydecomposition methods [18] to get a better view of

the nature of the bonding.

In the scheme used here occupation of virtual

orbitals is rigorously zero since the effective MOs

of Eq. (4) have been generated from the KS orbi-

tals by a unitary transformation (the product of U

of Eq. (2) and T of Eq. (4)) and therefore form a

complete and orthogonal basis for the space ofoccupied orbitals. Completeness is a distinctive

advantage for the purpose of decomposition of the

density of states. The projection of the density of

states on an orbital /IcðrÞ

DIcð�Þ ¼Xk

hwkj/Ici�� 2dð�� kÞ ð6Þ

is normalized to unity when integrated over one-

electron energiesZd�DIcð�Þ ¼

Xk

hwkj/Ici�� 2 ¼ 1: ð7Þ

Furthermore, the first moment yields the orbital

energyZd��DIcð�Þ ¼

Xk

�k hwkj/Ici�� 2 ¼ h/IcjHHsj/Ici ¼ aIc:

ð8Þ

Similarly the expectation value of the square of

the one-electron hamiltonian (using the mean en-

ergy as reference)

b2Ic ¼ h/IcjHH 2s j/Ici � h/IcjHHsj/Ici

2 ð9Þ

can be identified with the squared width of the

projected density of states

b2Ic ¼Zd� ��

� aIc

�2DIcð�Þ: ð10Þ

Substitution in Eq. (9) of the resolution of the

identity using Wannier states gives (recall virtual

states play no role in the present approach):

b2Ic ¼XNJk

h/IcjHHsj/Jki�� 2 � h/IcjHHsj/Ici

2

¼XNJ 6¼I

Xk

h/IcjHHsj/Jki�� 2; ð11Þ

where N is the number of molecules. From this weconclude that b2Ic can be regarded as a measure ofthe electronic interaction of molecule I with thesurrounding solvent. This interpretation is in line

with the projected density of states picture as used

in tight binding models in solid state physics [3].

There is also a parallel to the resonance integrals

in H€uuckel theory (from which our notation is

borrowed).

5. Thermally averaged bands

Thermal disorder creates a different environ-

ment for each molecule. As a result there will be

some spread of the effective MO energies around a

mean value

ac ¼1

N

XNI

aIc: ð12Þ

The magnitude of fluctuations in the orbital

energies is measured by the corresponding vari-

ance

D2c ¼1

N

XNI

a2Ic

" #� 1

N

XNI

aIc

" #2: ð13Þ

The thermal spread D2c is a form of inhomoge-neous broadening which should be distinguished


from the electronic dispersion quantified by the

parameter b2Ic (see Eqs. (9) and (11)). In order toassess the relative importance of these sources of

broadening for the band width of the total density

of states (Fig. 1) we define an average density of

states for each band c

Dcð�Þ ¼1

N

XNI

DIcð�Þ: ð14Þ

Owing to Eq. (7) the total density of states (Eq.

(1)) is a superposition of these overlapping average

band densities

Dð�Þ ¼ NX

c

Dcð�Þ: ð15Þ

The center of the band defined by Dc coincides

with the average orbital energyZd��Dcð�Þ ¼ ac: ð16Þ

The second moment of average MO bands

lð2Þc ¼

Zd� ��

� ac

�2Dcð�Þ ð17Þ

is related to the electronic and thermal broadening

as

lð2Þc ¼ b2c þ D2c ; ð18Þ

where the average electronic interaction b2c is de-fined as

b2c ¼1

N

XNI

b2Ic: ð19Þ

The square additivity property of Eq. (18) is

rigorous and is easily demonstrated by substituting

Eqs. (9) and (13). It is the central relation in our

analysis.

Table 1

Average band position ac (Eq. (12)) and width parameters in eV for

MO a D

1A1 )25.66 0.43

1B2 )13.15 0.46

2A1 )10.24 0.48

1B1 )8.26 0.42

Listed are the square root of the statistical variance D2c of orbitalsum and the second moment lð2Þ

c of the average projected density of

6. Results and discussion

Effective MOs were generated for the same 20

configurations used for the computation of the total

density of states (see Section 2). The effective MOenergies on a specific molecule were then energy

ordered to determine which band they belonged to.

The three parameters ac, Dc and bc were calculated

for the four effective MOs 1A1; 1B2; 2A1 and 1B1.

The double average over time and molecules can be

conveniently carried out as a single average over 20

times the number of molecules in a single configu-

ration (640). These values are presented in Table 1.The corresponding band densities of states Dcð�Þ(Eq. (14)) are compared to the total density of

states in Fig. 2. The second moments lð2Þc have

been calculated and are also presented in Table 1.

Square additivity (Eq. (18)) is satisfied for all

bands (deviations are due to numerical or statistical

uncertainties).

The statistical broadening Dc is about 0.45 eVfor all bands independent of the character or po-

sition of the band. For a comparison, the vibra-

tional thermal modulation as determined by an ab

initio molecular dynamics run of a single molecule

in vacuum at 300 K, gives second moments of

0.09, 0.17, 0.13 and 0.02 eV for respectively the

1A1; 1B2; 2A1 and 1B1 levels. This observation

confirms that the variation in the effective orbitalenergy is largely due to fluctuations in the local

electrostatic potential shifting the orbitals by ap-

proximately the same amount. In contrast, the

electronic broadening bc is rather sensitive to

the nature of the orbital. Whereas for the 1A1 MO

the electronic broadening of 0.6 eV is compara-

ble to the thermal broadening, the bc � 1:3 eVresult for the three valence bands 1B2; 2A1; 1B1 is

the effective MO bands in liquid water

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ b2

q ffiffiffiffiffiffiffilð2Þ

p0.58 0.86 0.87

1.41 1.25 1.27

1.57 1.31 1.34

1.11 1.10 1.13

energy (Eq. (13)), the electronic broadening b2c (Eq. (19)), theirstates (Eq. (17)).

Fig. 2. Decomposition of total density of states (dashed curves)

in projected densities of states (solid curves) for the three va-

lence orbitals of water (see Eq. (15)). Shown are the densities

NDcð�Þ). The projection on the 1A1 orbital (not shown) is

identical to the corresponding peak in the total density of states

(see Fig. 1).


significantly larger. Moreover, there is also a slight

variation between the valence bands. The elec-

tronic broadening of the HOMO is less than that

of the 2A1 and 1B2 MOs.A thermal width of approximately 0.5 eV seems

reasonable in view of the 1 eV fluctuations in in-

teraction energies observed for aqueous systems.

More surprising is the comparatively large elec-

tronic broadening. For an assessment of this result

we note that the band densities of the three inter-

acting orbitals 1B2; 2A1; 1B1 each exhibit long

tails strongly overlapping with the other two peaksin the three state manifold (Fig. 2). These tails add

substantially to the second moment. Had they not

been there the electronic broadening would have

come out smaller and in fact similar to the thermal

broadening. This could be a manifestation of

orbital interaction in hydrogen bonded systems.

Alternatively, the presence of these tails could be

interpreted as an indication that our scheme has

achieved only a partial disentanglement of the

overlapping bands.

To resolve these issues further technical devel-

opments will be needed, for example along thelines of [19]. In this approach, which is based on

the original reciprocal space implementation of

maximally localized Wannier functions [4], the

band states in a specified energy window are re-

solved in subbands by imposing restrictions on the

energy dispersion in reciprocal space. Whether the

improved energy separation is compatable with

the molecular nature of the states is an interestingquestion. Finally, the band tails could also be

specific to the Wannier scheme for the construc-

tion of localized molecular orbitals. This could be

investigated by comparing to the results of the

application of a more traditional fragment based

scheme, such as outlined in [17], which takes

coupling to empty orbitals into account.

7. Summary and conclusion

It would be a significant advancement to obtain

effective molecular orbitals in solution which can

then be used in the interpretation of intramolecular

bonding and for describing intermolecular interac-

tions. For example, effective MOs for individualmolecules in solution would enable us to study sol-

ute reactivity independently from (but still affected

by) the surrounding solvent. The focus in the pres-

ent study was on the computation of the width of

such effective states. It was found that for liquid

water the thermal and electronic broadening of the

effective MO levels is of the order of 0.5–1 eV,

bringing their width in the range of the energy gapsbetween frontier orbitals of reactive molecules.

A second conclusion is that thermal and elec-

tronic broadening are comparable in magnitude,

with the electronic interactions dominating. When

interpreted in terms of the tight binding models

for the electronic states of disordered solids, such

as the Anderson model, this observation has

implications also for the spatial extent of thecanonical (KS) orbitals. A ratio of 0.5–1 between

the statistical spread and electronic band width, as


computed here for water, suggests that a sizable

fraction of the KS states at the edges of the bands

could be localized. This set of states will include

the HOMO, which may have consequences for,

in particular, redox reactions. We also noted a

number of ambiguities in our localization schemerequiring further development. In our view, how-

ever, considerations concerning the electronic

width of energy levels are important and may also

impact on, for example, the validity and accuracy

of QM/MM methods.

Acknowledgements

We are grateful for the support received by PH

from the Leverhulme Trust (Grant F/09 650/B).

Part of the computations were carried out using

the resources of the High Performance Computing

Facility of University of Cambridge.

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Thermal versus electronic broadening in the density of states of liquid water