Superexchange, Localized, and Domain-Localized ... - ACS Publications

J. Phys. Chem. B 2001, 105, 2153-2162

2153

Superexchange, Localized, and Domain-Localized Charge States for Intramolecular Electron Transfer in Large Molecules and in Arrays of Quantum Dots F. Remacle† De´ partement de Chimie, B6, UniVersite´ de Lie` ge, B 4000 Lie` ge, Belgium

R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew UniVersity, Jerusalem 91904, Israel, and Department of Chemistry and Biochemistry, UniVersity of California Los Angeles, Los Angeles, California 90095 ReceiVed: August 16, 2000

Superexchange is a longer-range electron-transfer mediated by a nonresonant bridge between the donating and accepting states. We discuss a coupled set of donor/acceptor levels that are not resonant, with special reference to coupling of intermediate strengths. Examples of such systems are peptide cations or arrays of quantum dots. If the coupling is strong enough to overcome the gaps, charge can migrate. If the coupling is too weak, the charge remains localized. In the intermediate case, the charge is shown to be localized over a finite, connected, subset of sites. Degenerate perturbation theory provides a suitable zero-order basis for this intermediate regime. In a time dependent language, in the domain-localized regime, the charge migrates over a limited range of states. Also discussed is an effect of electron correlation, the so-called Coulomb blockade, on charge localization with computational examples. The experimental probing of the domain-localized regime is considered. Probes of the energy dependence of the local density of states such as scanning tunneling microscopy (STM) of arrays of quantum dots and photoelectron spectroscopy (PES) of chromophore bearing molecules are suggested.

I. Introduction Intramolecular electron transfer is often discussed between a donor and acceptor that are separated by a bridge.1-4 The question is then raised if, to any significant extent, the charge is also to be found on the bridge. If the levels of the bridge are too high compared to its end points, then the charge is to be found mostly at either end and the role of the bridge is to allow the coupling to be effectively longer ranged. The term “superexchange”5,6 is usually applied to such a situation. Here we consider a set of coupled units where the energies of the neighboring units can differ in either direction. The units can be, for example, amino acid residues in an isolated peptide cation7,8 or quantum dots9-12 assembled in an array. The units carry orbitals that can accept or donate an electron, and the orbital energies can be thought of as the ionization potentials of the uncoupled units. These energies are obviously different for different amino acids although for most amino acids they differ by less than 0.5 eV. For quantum dots the origin of the difference is that the dots are seldom truly identical. The nature and extent of the variation depends on the method of preparation but it is often above 5% when the dots are prepared by wet chemical methods. The energies of the valence orbital(s) depend on the size of the dots and therefore they will fluctuate. Electron transfer is possible because the units are coupled.13 The strength of the coupling can sometimes be tuned. For an ordered two-dimensional array of quantum dots it can be controlled by compressing the array.14 In a chain molecule1 or in a peptide,15 the magnitude of the coupling of the residues will be governed by the relative alignment of the units.16 It is †

Chercheur Qualifie´, FNRS, Belgium. * Corresponding author. Fax: 972-2-6513742; e-mail: [email protected].

to be expected, and we will discuss this in detail, that, for our purpose, the effective coupling between two units is their electronic coupling scaled by the gap in their energies. Therefore, another way of tuning the coupling is by varying the nature of the units. One can realistically think of different sequences of amino acids along the peptide7 or of arrays of quantum dots of different compositions.17,18 The coupled units can also be atoms in a large molecule (e.g., refs 19 and 20) with the difference, further discussed below, that the charging energy of atoms is typically much higher. When the effective coupling is increased, one can demonstrate a transition from a localized to a delocalized electronic state. In two-dimensional monolayers of quantum dots,21 the coupling can be tuned by compression of the array. The resulting changes in the electronic structure can be probed by nonlinear spectroscopy14,22 or by measuring the frequency dependent dielectric constant 23,24 or by measuring the STM current.25,26 Either measurement exhibits a transition that can be interpreted as due to delocalization of the electronic states for effectively coupled dots.27 In peptides, the measurement is indirect and charge delocalization is monitored through its chemical effects. Specifically, by using a chromophore, the peptide chain is ionized at one end and charge migration is probed by measuring the resulting fragmentation pattern. When charge migration is possible, the chain can break selectively at a site far from the site of the local ionization. Charge will flow when the gaps between the energies of the amino acid residues can be bridged by the electronic coupling. So that if charge migration is hindered by the substitution of an amino acid residue along the chain, the fragmentation occurs at the site of the local ionization.7,8

10.1021/jp002972z CCC: $20.00 © 2001 American Chemical Society Published on Web 02/15/2001

2154 J. Phys. Chem. B, Vol. 105, No. 11, 2001

Remacle and Levine

The systems discussed above are different, as are the probes for charge migration. Yet they both exhibit two clear limiting behaviors: (i) In effective coupling, the electronic states are delocalized. This requires that the amplitude β for charge transfer can couple nonresonant neighboring sites, β > ∆R, where ∆R is the difference in the energies (the IP’s). (ii) In the ineffective coupling regime, β < ∆R, the electronic states are localized. Of course, the electronic states are strictly localized only when the coupling is negligible, β/∆R f 0, and what we mean by “localized” is that the amplitude to find the charge decreases exponentially as one moves away from the charge carrying unit. It is this decrease that is manifested in the exponential decrease of the electron-transfer probability with distance. The question we address in this paper is if there is an intermediate regime that is qualitatively different from either limit. A regime where the charge is neither localized nor delocalized. We choose to give a name to this regime because it can be experimentally characterized and corresponds to a physical situation different from the fully localized regime. We will argue that there is such a regime and that it is “domain localized”. In this regime the charge is localized on a number of geometrically adjacent participating units, the size of the domain, with the characteristic that the domain is bigger than one unit but smaller than the size, N, of the system

N > Nparticipating > 1

(1.1)

Of course, domains can only be usefully defined if the system is large enough, N . 1. It is the geometrical connectivity of the domain that distinguishes it from a superexchange limit. In section VI we discuss in detail why for either peptides or assemblies of quantum dots it is necessary to examine the effects of electron correlation. We do so by going beyond the oneelectron Hu¨ckel approximation and we use a many-electron description of the electronic structure. Technically, we implement a full configuration interaction of the many-electron states of a given total multiplicity. N ) 19 is the smallest planar hexagonal array of two shells about a central unit, 19 ) 1 + 6 + 12. A full configuration interaction for an array of 19 quantum dots requires almost 3 billion (doublet) electronic states. Computationally, diagonalizing a realistic Hamiltonian, which allows for electron-electron repulsion, for a large system is therefore not tractable and the results of the many-electron description are limited to N ) 7, the simplest hexagonal array. For larger arrays, we use a Hu¨ckel type, independent electrons, Hamiltonian for which N can be taken to be quite large. For the Hu¨ckel Hamiltonian one can examine the N molecular orbitals (ψµ, µ ) 1, 2, ..., N) directly since they are obtained as a linear combination of orbitals localized over the sites N

ψµ )

cµiφi ∑ i)1

(1.2)

where |cµi|2 is the weight of the ith site. For a delocalized molecular orbital the coefficients are all about equal (up to quantal fluctuations28-30). Normalization of the molecular orbital then implies that for orthogonal site orbitals the typical value is |cµi|2 ) 1/N. For a localized molecular orbital, one particular weight is about unity and all the others are exponentially small. For an N site system one can therefore define the number of sites that participate in the µth molecular orbital as31 N

Nparticipating µ ) 1/

|cµi|4 ∑ i)1

(1.3)

The molecular orbital µ is domain localized if (i) Nparticipatingµ satisfies eq 1 and (ii) those sites that make a large contribution to the orbital are geometrically neighboring sites. In section VII we discuss a generalization of the measure (1.3) to many-electron states, a generalization that is needed in order to have a definition valid for strongly correlated electrons. It turns out that the generalization also points out a way to experimentally probe the magnitude of the number of participating sites. The Hamiltonian is defined in section II. The transformation to a domain-localized Hamiltonian and orbitals is discussed in section III and numerical results are presented in section IV. In section V it is pointed out that the discussion can also be carried out in a time dependent language. First we note that even when the exchange coupling is small, as long as it is finite then, eventually, the charge will sample the entire system. But on a shorter time scale the picture is different. Charge initially placed on a site will not, in the localized regime, move away any time soon. In the domain-localized regime the charge will sample the entire domain but for a long while will not move outside of it. There will be a clear separation in time scales between the sampling of the domain and the sampling of the system. The charge migration will rapidly sample the entire system if the wave function is delocalized. Domain localization occurs when the coupling between the sites is not strong. The effective strength of the coupling, as discussed in section III, is governed by two factors, the gap in energies between adjacent sites and the site-site coupling that moves the charge from one site to another. Either factor can be subject to experimental control, the gaps by the preparation of the system and the coupling by geometrical changes. Beginning with section VI we discuss yet another aspect that is relevant to the magnitude of the coupling. This is the role of electron correlation. To understand why it is at all important, consider a reduction in the strength of the site-site coupling by extending the distance. At some point, the molecular orbital description will then qualitatively fail. This shortcoming is long known.32 For the simple example of a homonuclear diatomic molecule, X2, as the X-X distance increases the covalent X:X and ionic X+X- configurations have equal weight. Ionic states are stabilized by the Coulomb attraction but are typically asymptotically at a higher energy because it takes more energy to ionize X than is recovered by forming X-. To properly discriminate against the ionic states one must, at least, allow for the Coulombic repulsion between two electrons that are on the same site. This refinement (known as “the charging energy”) is introduced in section VI. It is necessary in order to ensure that as we increase the intersite distances, the wave function has the correct asymptotic form. In particular, for the units of interest to us, say quantum dots or amino acid residues, the charging energy is much lower than for most atoms. The discussion of the homonuclear diatomic can be refined by considering also the heteronuclear one, X-Y, where the valence orbital energies of the two atoms are not the same. Here the weights of the two distinct ionic and the covalent states will depend on the X-Y separation and asymptotically the ground state will converge to a particular state, typically the covalent one. But here too, when the X-Y separation is large and the coupling is therefore weak, allowing for the role of the charging energy will have a marked quantitative effect on the weight of the different configurations. The question of experimental probes for the transition from localized to domain localized to delocalized charge distribution is discussed in section VII.

Electron Transfer in Large Molecules

J. Phys. Chem. B, Vol. 105, No. 11, 2001 2155

In this paper we center attention on the sites on which the charge is localized or, in a time dependent language, the sites that the migrating charge can access. Our studies of the classical limit of electronic degrees of freedom33 suggest that another aspect of localization is that of the overlap charge density. (The terminology is that of Slater.34 This is known as the bond order in the MO theory or as the off-diagonal elements of the one electron density matrix6). We are in the process of a closer look at this point. Another needed extension is that we have only considered pure electronic states. In thermal systems one must allow for mixed states. II. Hamiltonian A realistic Hamiltonian when there is one valence orbital per site is that of Pariser-Parr-Pople, PPP, or a simpler version, used originally in solid-state physics, due to Hubbard6,35

H ) H 0 + H1

(2.1)

Here H0 is a one spin-orbital per site, N site, Hamiltonian of the tight binding (or Hu¨ckel) type, N

H0 )

2

∑i ∑σ Ri

N

∑ i,j

† ai,σ ai,σ +

∑σ ai,σ† aj,σ

βij

(2.2)

near neighbors

The a† and a in eq 2.2 are the creation and annihilation operators for an electron at a given site and with a given spin (σ ) up or down). The sites need not have equal energies (IP’s) and the variations in the site energies Ri are discussed below. β, the transfer integral, is the coupling that moves an electron from one orbital to another. Note that because of the spin labels, an electron can only move into an empty spin-orbital. The two-electron part of the Hamiltonian is

H1 )

1

∑IiEˆ ii(Eˆ ii - 1) 2 i on site Coulomb repulsion

+

1 2

′γijEˆ iiEˆ jj ∑ i,j

(2.3)

cross site Coulomb repulsion

2


i, j ) 1, ..., N

III. Domain-Localized Basis It is straightforward to write zero-order Hamiltonians that correspond to the localized and delocalized limits. The eigenstates of these define a basis suitable for either limit. In this section we derive a basis suitable for the domain-localized regime. Delocalized orbitals are eigenstates of the Hu¨ckel type Hamiltonian N

2

N

∑i ∑σ ai,σ† ai,σ + β

∑ i,j

Hband ) R j

2


(3.1)

near neighbors

In (2.3) we introduced the notation of the unitary group36,37

Eˆ ij ≡

is similar to the orbital of a particle in a spherical box of radius R. If the potential outside is infinite, the energy can be readily computed analytically and scales as R-2. The radius of the dot determines therefore the energy of the highest occupied orbital and also the charging energy. Fluctuations in the size of the dots imply fluctuations in both the energy of the highest occupied orbital of the dot, |δR| ) R2|δR/R| and in the charging energy |δI| ) I|δR/R|. The fractional fluctuations are roughly the same because both energies decrease with increasing size of the dot. In the computations we draw a magnitude of R from a narrow size distribution and use it to compute the value of the energy parameters of that dot. For the peptides, we use the local IP’s, as specified by the composition. In the Hu¨ckel approximation, computing the molecular orbitals means diagonalizing an N by N symmetric Hamiltonian matrix. When the electron repulsion terms are included, a simple orbital picture is no longer valid and one needs to use manyelectron wave functions. If these are expressed as a linear combination of orbital wave functions, the size of the basis increases as a high power of the number, N, of sites. This is because of the many ionic states that are possible. The unitary group approach allows one to construct the Hamiltonian matrix in a very systematic fashion,36,37 but by N ≈ 10 the sheer size of the matrix presents a problem if one wants to obtain also the excited states.

(2.4)

where Eˆ ii is the number operator for electrons on the site i, while Eˆ ij shifts charge from site j to site i. The first, sometimes called Hubbard,38 term in (2.3) is the electrostatic repulsion between two electrons (of opposite spins) that are on the same site. The second term is the Coulomb repulsion of electrons on different sites. We will include the two-electron, H1, part, of the Hamiltonian only from section VI on. The reason is that, when the exchange coupling β is large, the role of H1 can, to leading order, be neglected. For the systems we are interested in, the size of the units on each site makes the charging energy low relative to what it is in atoms. There are two simple ways to estimate it. One35,39 is in terms of the energy cost of the process A + A f A+ + A-. The other is by introducing the (size dependent) capacity C of the subunit, I ) e2/C. For the simple case of a spherical subunit C ∝ R, where R is the radius of the subunit. R, β, γ and I carry labels of the sites because the sites are not necessarily equivalent. This is due to the fluctuations in size of the dots or the difference in the amino acid residues in a peptide. Taking the dots as an example, their valence orbital

The states of this Hamiltonian are resonantly coupled as they have a common energy, R j , which is obtained by averaging over the site variations. For either component of the spin σ, the Hamiltonian matrix for this limit has the familiar form Hband ) R j I + βM, where M is the, so-called, N × N “topological” or “adjacency” matrix,40 i.e., a matrix having a unit entry whenever the column and row indices denote near neighbors. The eigenvectors of Hband ()eigenvectors of M) are delocalized over the entire set of connected units and the eigenvalues are R j + βm, where m is an eigenvalue of M. The key point is that the eigenstates of Hband are delocalized for any nonzero value of β. The subscript “band” is a reference to this banded spectrum, familiar from π electron theory of aromatic molecules.35,40 The zero-order Hamiltonian whose eigenvectors are localized is obtained from (2.1) when the sites are uncoupled. We take the β f 0 limit by increasing the distance between the units, so that the cross Coulomb term can also be neglected βf0

H 98 Hsite )

N

1

N

RiEˆ ii + ∑IiEˆ ii(Eˆ ii - 1) ∑ 2 i)1 i)1

(3.2)

The eigenvalues can be immediately written down because all the operators are diagonal when the basis is that of site-localized orbitals. Using roman letters for site states

2156 J. Phys. Chem. B, Vol. 105, No. 11, 2001 N

Esite m )

1

Remacle and Levine

N

Rinim + ∑Iinim(nim - 1) ∑ 2 i)1 i)1

(3.3)

where nim is the occupancy of site i in the mth site state. In the Hu¨ckel limit only the first term in (3.2) is retained N

Hsite )

RiEˆ ii ∑ i)1

(3.4)

The essential difference between the band and site Hamiltonians is that in the site Hamiltonian the sites are not necessarily resonant. On the other hand, the variations in the site energies can be small. Our purpose is to allow coupling primarily between such sites where β can bridge the gap between the site energies. The starting point is the full Hu¨ckel Hamiltonian N

HHu¨ckel )

∑ i)1

N

RiEˆ ii + β

∑ i,j

Eˆ ij

(3.5)

near neighbors

which clearly reduces to the site Hamiltonian (3.4) when there is no coupling and to the band Hamiltonian (3.1) when the sites are all equivalent. We then subject it to a unitary transformation

H ˜ ≡ exp(-iβSˆ )HHu¨ckel exp(iβSˆ )

(3.6)

and choose the operator Sˆ such that, in the site basis, the transformed Hamiltonian has no coupling term linear in β. This is a Van Vleck type transformation.41 Using square brackets to denote a commutator, the term in H ˜ which is linear in β is HHu¨ckel - Hsite + i[Hsite, Sˆ ]. The condition that this term is diagonal serves to define Sˆ in the site basis and thereby the domainlocalized states

Sij ≡ iMij/(Ri - Rj)

(3.7)

where the matrix M is the adjacency matrix. Equation 3.8

φ˜ ≡ exp(iβSˆ )φi ) (1 + iβSˆ - β2Sˆ 2/2 - ...)φi

(3.8)

defines the transformation to “domain-localized states”. It shows that a domain centered over site i is really a domain; that is, it includes other sites with a nonexponentially small weight, when the matrix elements of βS in the site basis are not small compared to unity. The matrix elements of S are not quite identical for different sites and so we use the following definition of an effective coupling constant, β*. The matrix S has no diagonal elements. Therefore, the jth eigenvalue of S is bounded by ∑i*j|Sij|. (See for example42 theorem 7.5.4). We next average this bound over all the sites, so that

β* )

∑j (∑i βij/|Ri - Rj|)/∑j 1 ≡ 〈∑i βij/|Ri - Rj|〉

(3.9)

Note that for an array of dots, while the site energies are essentially not correlated and so the site energies are randomly j so that 〈Ri - Rj〉 ) 0, the average in (3.9) fluctuating, 〈Ri〉 ) R does not vanish because the contribution of each site is only due to those other sites to which it is coupled. Nor will such an average vanish for other systems such as a peptide, again because of the restriction on the summation in (3.9). A quick estimate for the effective coupling constant is provided by the argument that a typical difference in two site energies is ∆R/N, where N is the number of sites whose energies lie within an interval ∆R. Therefore, one has that β* ≈ Nβ/∆R. We have

verified that the value of β* computed by evaluating (3.9) and averaging over several samplings differs by at most a few percent from the simple estimate. The considerations of this section remain relevant also in the more general case, when the Coulomb repulsion between electrons is explicitly included. We will only provide a full discussion of this case beginning with section VI, but it is convenient to note here certain results that are relevant. The key point is that the localized limit is defined by the site Hamiltonian. In the Hu¨ckel limit this is given by eq 3.4 and the eigenstates are the site orbitals. In the more general case the site Hamiltonian is given by eq 3.2 and can be systematically be constructed by a unitary group formalism.43 As can be seen from (3.2), the eigenstates are those many-electron states where each site is occupied by a definite number, 0, 1, or 2, of electrons. These states are therefore eigenstates of the number operators Eˆ ii, i ) 1, 2, ..., n and hence eigenstates of the site Hamiltonian. Computationally, these are the basis states that we use to construct the matrix of the full Hamiltonian, eq 2.1. The computational advantage is that these states diagonalize the two nontrivial terms that are referred to as the H1, equation (2.3). These represent the electrostatic effects. The only off-diagonal terms in that basis are due to the site-site exchange coupling β, just as in the Hu¨ckel level description. The site many-electron basis states are quasi-degenerate because, when each site contributes just one electron, there can be many permutations of the occupancies which will all give rise to nearly equal site energies, eq 3.3. The transformation to a domain-localized basis can therefore to be applied as at the Hu¨ckel level. The only difference is in the indices, since (3.7) is to be replaced by site Smn ≡ iβmn/(Esite m - En )

(3.10)

In (3.10), the indices range over the number of site basis states. Computing the coupling between site states has been discussed in detail before43 and lends itself to a very systematic approach. Finally, eq 3.9 for the effective coupling strength is to be site replaced by 〈βmn/|Esite m - En |〉. In considering the effective coupling we have to distinguish two cases. These are characterized by another dimensionless parameter. It is convenient to think of it in general as β/I where I is the charging energy, but for our immediate purpose it is the measure m* of the ionic character

m* ≡ 〈I Mij/|Ri - Rj|〉 ) (I/β)β*

(3.11)

If m* > 1 it is energetically more favorable to keep the electrons as much as possible on different sites. The set of site states can then be represented as a series of energetically nonoverlapping bands. For the case of N electrons on N sites, the lowest band will be the one where each electron is on its distinct site. We call these states “covalent” because no site is empty or is doubly occupied and the electron spins are such that it is a doublet. N From eq 3.3 Esite j so that all the covalent states m ) ∑i)1Ri ) NR are degenerate. The next higher band of states has one site empty and one doubly occupied. This lowest ionic band has states N spread out: Esite j - Rj + Rk + m ) ∑i)1Ri - Rj + Rk + I ) NR I, where in the many-electron state m, sites j and k are empty and doubly occupied, respectively. The mean energy of the lowest ionic band is clearly NR j + I, which is higher by the energy I than the lowest, covalent band. Its width is ∆R and the bands are not overlapping since, by assumption, I > ∆R. The next conduction band has two sites empty and two doubly occupied. Its mean energy is NR j + 2I, etc. A schematic



representation of the four bands that are possible for a hexagonal array of seven sites is shown in eq 3.12. The number of states

within each band and the mean energy (as computed above) are indicated. The combinatorial counting of the number of possible states has been discussed by Paldus.37 If m* > 1 the site Hamiltonian has a series of nonoverlapping bands of localized states. The states within each band are quasi-degenerate since the bandwidth is ∆R but the bands are distinct. If m* > 1 it will take a not small value of the site-site coupling β before the onset of domain localization. This is because the site-site coupling needs to bridge the gap between nonoverlapping bands. If m* < 1, the different bands of the site Hamiltonian are overlapping. All the localized states are then members of one band and all localized states are potentially quasi-degenerate. Once we switch on the site-site coupling, even low values of β can couple localized states. The scope for domain-localized states is far wider. The same conclusions follow from eq 3.11. If m* > 1 it requires that β > I for effective coupling. If m* < 1 even when β < I, the coupling can still be effective enough to delocalize states over a finite domain. IV. Computational Results The most direct visual evidence for a domain-localized regime is a plot of the weights of different sites in a molecular orbital, as a function of the effective coupling. Figure 1 is such a plot for a hexagonal array of 91 (four shells) quantum dots. The parameters are those appropriate to Ag nanodots of 3 nm.24 For such an array we have determined the dependence of the coupling β on the distance D between dots, by fitting to the measured second harmonic generation spectra22 using the coupling D >D0

β ) (β0/2)(1 + tanh[(D0 - D)/4LR]) 98 (β0/2) exp(-D/2LR) (4.1) This dependence on D/2R, and particularly the decrease as exp(-D/2LR), was found to satisfactorily account for the measured frequency response of the dielectric constant.24 The scale parameter L was found to have the value L ) 1/11. Note that this is a fairly steep decline with D/2R. The upper scale in Figure 1 is expressed in terms of the distance D given in units of the mean dot diameter. The range 1 < D/2R < 2 is experimentally accessible using a Langmuir-Blodgett arrangement.14 Therefore, Figure 1 suggests that the domain-localized regime can be experimentally accessed.23,25 The results shown in Figure 1 are computed by drawing, once, a set of site energies for a 5% variation in size. Different coupling strengths are obtained by varying the magnitude of β. It is a Hu¨ckel level computation. It is seen in Figure 1 that the weights of the sites change from an average of 1/N at strong coupling to 1/Nparticipating, where Nparticipating < N at weaker coupling and decrease to order unity at very weak coupling. To emphasize that domain-localized states are geometrically localized, Figure 2 shows the distribution of the weights of the individual sites over the entire array.

Figure 1. Weights |cµi|2 of two molecular orbitals (MO’s), m ) 3 and 7, respectively, as a function of the effective coupling Nβ/∆R, lower abscissa, or of the compression of the array, upper abscissa, for various values of the site index i. Computed for a hexagonal array of N ) 91 sites (see Figure 2) in the Hu¨ckel approximation. The range ∆R of the fluctuation in the site energies is defined as follows: The site energies are sampled as Ri ) R j (1 + δRi) ≡ R j [1 + δR(rani - 0.5)] where rani is a random number in the range [0, 1] and ∆R ) R j δR. For the computation shown here, R j is 10 β0 and there are 5% fluctuation in the site energies. The sites around which the two MO’s eventually localize are identified in the plot. The distance dependence of the coupling β is given by eq 4.1 where D is the distance between neighboring sites and R is the radius of a site. In the strong coupling, delocalized, limit, |cµi|2 ≈ 1/91 up to quantal fluctuations. In the domainlocalized regime |cµi|2 ≈ 1/Nparticipating, where the number of participating sites is significantly smaller than N ) 91. See also Figures 2 and 3. The domain-localized regime is seen to be centered about β* ≈ 1. See the text after eq 3.9 for the reason for the estimate β* ≈ Nβ/∆R.

Figure 3 shows a histogram of the number of molecular orbitals with a given participation number of site states, cf. eq 1.3, for different values of the coupling parameter β* ) Nβ/ ∆R or, equivalently, of the compression D/2R. The computation is for a 91 site array and it verifies the qualitative impression conveyed by Figure 2: In the domain-localized regime 60% of the 91 molecular orbitals are localized over 1-5 sites, 38% are localized over 6-10 sites, and 2 are more delocalized. The elements of the transformation matrix S to the domainlocalized basis, cf. eq 3.7, are shown in Figure 4 for the same site energies as in Figure 3. It is seen that a particular site is either effectively coupled to a few other sites or is effectively isolated. V. Charge Migration The discussion so far has been in terms of the stationary states. Once the Hamiltonian is diagonalized one can, of course,


Remacle and Levine

Figure 3. Histogram of the number of molecular orbitals whose number of participating states is in a given range, as shown. Computed for a 91-site array as in Figures 1 and 2. The number of participating states is computed from its definition, eq 1.3. In the domain-localized regime, D/2R ) 1.6, of the 91 molecular orbitals 89 have between 1 and 10 participating sites. Of these, 54 MO’s have between 1 and 5 participating sites.

Figure 2. Hexagonal lattice of N ) 91 sites with a representation of the weight of the lowest molecular orbital shown on each site. The plot is to demonstrate that in the domain-localized regime, bottom panel, the significantly participating sites indeed form a geometrical domain. The shading indicates the range for the bins of the histogram. Blank sites are almost empty, |cµi|2 e 1/91. Light gray sites have a typical weight 1/91 e |cµi|2 e 4/91. Dark gray sites are occupied well above the average 4/91 e |cµi|2 e 9/91. Black sites are strongly localized, 9/91 e |cµi|2. The delocalized behavior, top panel, is for a lattice compressed to D/2R ) 1.2. The domain-localized plot is for D/2R ) 1.6. As in Figure 1, 5% disorder in the site energies is included in the computations.

describe the time evolution of any initial state φ by expanding it in terms of the eigenstates whose time evolution is diagonal N

φ(t) )

∑

N

dµ exp(-iµt/p)ψµ )

µ)1

N

∑ ∑cµidµ exp(-iµt/p) i)1 µ)1 φi

(5.1)

As shown, the expansion can be rewritten in terms of site states and, if desired, also in terms of the domain-localized states, cf. eq 3.8 N

φ(t) )

N

N

(T†)ijcµjdµ exp(-iµt/p) ∑ ∑ ∑ i)1 j)1 µ)1 φ˜ i

(5.2)

In the localized regime it follows that a charge initially placed on a given site will hardly move whereas at stronger coupling the charge will migrate over all states of the domain. This can be seen analytically by choosing the weights dµ such that initially the state φ is site localized. Figure 5 is a graphical representation of these results. It shows the trajectory of the charge over a hexagonal array of 91 sites, in different coupling regimes. The plot is a parametric representation of the amplitude c(t) ) 〈φ(0)|φ(t)〉, where the initial state is localized on site 16. In the localized regime |c(t)| remains close to unity. In the domain-

localized regime |c(t)| rapidly decreases from its initial value of unity and remains confined between its initial value and 1/ xNparticipating ≈ 0.4 (Nparticipating ≈ 6). In the delocalized regime Nparticipating ≈ 36. The autocorrelation function 〈φ(0)|φ(t)〉 recurs much more rapidly in the domain-localized regime because the charge migrates over far fewer sites. Its Fourier transform will therefore extend to higher frequencies. VI. Role of Electron Correlation As discussed in the Introduction, one must allow for electron correlation as the coupling between the sites gets to be weak. The reason is that correlation between the electrons is brought in by the electrostatic repulsion terms in the Hamiltonian and the contribution from repulsion of two electrons on the same site will not diminish as the array expands. The lowering of the kinetic energy of the electrons drives delocalization. To lower the potential energy, electrons should be kept apart, i.e., on different sites. The correlation of electrons will therefore hinder delocalization. This will be particularly noticeable in the limit when all the sites are identical and therefore resonantly coupled. Once the Coulombic repulsion is explicitly incorporated, it requires a finite coupling strength between the sites before delocalization is possible. This is the celebrated Mott insulator to metal transition44,45 that is expected when β and I are comparable in magnitude. At the end of section III we obtained the same conclusions even when fluctuations in the site energies are allowed. Figure 6 shows the exactly computed excited-state energies for a planar array of seven sites, with electrostatic repulsion between electrons on the same site, for different amounts of variations in the site energies. The plot is vs the coupling strength, represented, via eq 4.1, in terms of the distance D between the centers of the quantum dots. The exact computations shown in Figure 6 bring forth a number of points. First of all, and as discussed in section III, at large separations, as the coupling becomes much smaller than


Figure 4. Elements of the matrix S, such that exp(iβS) generates the entire domain of coupled states, from one of its members. The matrix is defined by eq 3.7 where M is the (so-called, topological) matrix with unit entries for sites that are near neighbors and zeros otherwise. Explicitly, each panel shows Sij ) Mij/(Ri - Rj) for a given i and for 5% variation in the site energies, i.e., ∆R ) β0/2. Top panel: sites j that are effectively coupled to site i ) 16. Of the six near neighbors, two sites are effectively coupled and four others are weakly coupled. Bottom panel: sites j that are effectively coupled to site 27. Here the domain has 4 sites that are dominant.

the Coulomb energies, there are a number of ionic bands, which are at an energy I apart. The lowest ionic band has only one site doubly occupied. The band higher up has two sites doubly occupied, etc. In the absence of disorder, top panel, when all site energies are the same, the states of a band become degenerate as β f 0. The coupling can move the charge between different states of the same ionic band and so the ionic bands are broadened when the coupling strength is increased, e.g., by compression of the array. In the absence of disorder, the ionic bands merge when the intraband charge-transfer coupling becomes strong enough to overcome the Coulomb blockade. Variations in the energies of sites also broadens the band and so facilitates band mixing, as can be seen in Figure 6. In terms of the onset of delocalization, what Figure 6 shows is that there is a regime in D/2R where the Coulomb gap is already bridged, so that a simple MO approximation becomes realistic, yet where the states are not yet delocalized. In the ideal case of no disorder, when all site energies are equal and are resonant, the closing of the Coulomb gap is quite clear and sharp. It is a real transition, as is also shown in Figure 8 below. In detail, one sees that, as the site-site coupling is increased, the first Coulomb gap to close is that between the different ionic bands. As discussed above, this is because the coupling can move the charge between different states of the same ionic band and so the ionic bands broaden by compression of the array. In the ideal case of no disorder, the last gap to close is between


Figure 5. Time evolution of the amplitude of site 16, top panel of Figure 4, given that it was initially occupied. The plot is a parametric one where for successive values of time the imaginary value of the amplitude is plotted vs the real value. As time evolves, this generates the curve that is plotted. The results are shown for three coupling regimes. In the localized regime, light gray, D/2R ) 2, the weight of the state does not fall below about 0.999. In the domain-localized regime, heavy gray, D/2R ) 1.6, the weight of the state very rapidly drops from unity. But the charge is confined to the domain so that the weight does not fall below about 0.1, with an average value of 0.4 ≈ 1/xNparticipating. In the delocalized regime the weight rapidly drops to below 0.1 and stays in that range.

the lowest lying covalent band and the ionic bands. This is the Mott insulator to metal transition,44,45 mentioned earlier. Once variations in the site energies are included, the level structure is far less neatly ordered. This allows for another mechanism of broadening of the bands. When the variation in the site energies is limited, middle panel of Figure 6, the behavior at weak coupling is almost like in the ideal case, except that the ionic bands merge sooner. But once the site-site coupling begins to be comparable to the variation in the site energies, we enter the domain-localized regime. In this regime, because the charging energy is low, the bands have already merged. One can, of course, discuss also the case where β becomes comparable with variations in the site energies and yet β < I. For ordinary systems this is the norm and in this case one will have delocalization without a metallic behavior. For a more extensive disorder or any wider variation in the site energies, lower panel of Figure 6, the ionic and covalent bands are overlapping. This occurs because, if the variations in the site energies are comparable to the charging energy I, it can be energetically favorable to have an empty site and two electrons on another site, a site of low energy R. Because there is no Coulomb gap, the domain-localized regime can span a wider range. Another way to display the role of disorder is shown in Figure 7. The plot shows the expectation value 〈Eˆ ii〉 for each one of the seven sites, in the 40 lowest N electron states, arranged in order of increasing energy. The computation is for a value of the coupling which, for 5% disorder will give rise to domainlocalized states. In the absence of disorder the bands are therefore merged and one is beyond the Mott transition. Charging energy is included with the value I ) 0.3 eV, which is expected for Ag quantum dots of 3 nm.46 With increasing


Figure 6. Excitation spectrum for a hexagonal array of seven sites, computed, vs the lattice compression, with a Coulomb repulsion between electrons on the same site. There are 784 many-electron (doublet) states for such an array. The three panels show the spectra for no disorder in the site energies, a 5% disorder, and a considerable, 20% disorder, as indicated. β0 is the energy scale for the coupling strength, cf. eq 4.1 and R j ) 10β0. The top two panels show a low energy band of covalent states and a higher energy band of ionic states. When the disorder is large, there is extensive mixing of covalent and ionic states.26 Due to the site-site coupling, the spectrum shifts with the compression. The dashed lines show the second-order perturbation theory estimate for this shift. In the absence of disorder there is a clear merging of the covalent and ionic bands, known as the Mott transition.44,45 See Figure 7 for another view of these results.

disorder it is easier first to mix the ionic states between themselves so that the higher states show more variation in the values of 〈Eˆ ii〉. With more disorder, covalent and ionic states also become nearly resonant. Figures 6 and 7 suggest the following approach to the definition of a domain-localized regime when electron correlation is included. In the one-electron, Hu¨ckel, limit we asked for the charge on a site for a giVen molecular orbital. Once correlation is brought in the wave function is no longer a single (antisymmetrized) product of occupied orbitals. But for any wave function one has a one-electron (so-called, “reduced”)

Remacle and Levine

Figure 7. Charge on a site, for each one of the seven sites in the seven-site array, for each one of the first 40 (out of a total of 784) many-electron states. Each site is shown with a different marker but the ability to distinguish between the sites is not important. Same three panels as in Figure 6, computed for a value of the interdot coupling corresponding to D/2R ) 1.4. In the absence of disorder and just before the Mott transition, top panel, the charge is localized. In the domainlocalized regime, the low-energy spectrum is still primarily covalent but as the energy is increased it becomes more delocalized due to the participation of ionic states. For wide variations in the site energies, bottom panel, even the low-lying states are ionic.

density matrix.6 In the site orbital basis, the diagonal elements of this matrix are, up to normalization, the 〈Eˆ ii〉’s. So the manyelectron analogue of the orbital picture definition is to compute the 〈Eˆ ii〉’s for each energy eigenstate, as shown in Figure 7. Since the eigenstates are dense and form a band, we next examine a continuous version of such a plot. VII. Charge-Weighted Local Density of States One can also ask for the charge on a site for many-electron states in a given energy range. The corresponding quantity is the charge-weighted density of states:26

LDOSi(E) ≡ Tr[Eˆ iiδ(H ˆ - E)]

(7.1)

We refer to this quantity as the local (at site i) density of states



at the energy E. For actual computations one necessarily has to replace the delta function by including all the eigenstates whose energies are within a narrow finite interval [E - ∆E/2, E + ∆E/2] with the weight 1/∆E. With this understanding and using the notation of ref 47, we can write a practical version of (7.1) using a rectangle function II((E - H)/∆E) and putting a bar over E to denote the averaging over a small energy interval

LDOSi(E h ) ≡ Tr[Eˆ ii II((H ˆ - E)/∆E)]

(7.2)

Instead, one can define the density of states at a finite temperature T by summing over all states with a Gaussian weight centered about E with a width kT where k is Boltzmann’s constant. One way or another, eq 7.1 cannot be computed at a really sharp energy. There is a corresponding problem for the experimentalist who wants to measure the local density by an STM experiment.25,48 The energy resolution is finite. The rapid variation of the 〈Eˆ ii〉’s, Figure 7, means that the averaging over several states will result in a value nearer to unity. The primary h ) as E h changes will therefore be from variation of LDOSi(E changes in the number of energy eigenstates in a given energy interval ∆E. Figure 8 shows the local density of states, at low energies, for a series of quite similar lattice spacings, so as to show the sharp closing of the Coulomb gap, in the absence of disorder. As discussed above, this closure is the gateway to the domainlocalized regime. Figure 9 is for a wider span of coupling strengths, from the localized to the delocalized regimes and the computations include a 5% variation in the site energies. The local density of states can be measured by scanning tunneling spectroscopy.48-50 The point relevant to the present discussion is that the measured spectra can differ for different locations of the tip over the sample. Figures 8 and 9 are computed using the full Hamiltonian, eq 2.1, for a seven-site array. It is computationally prohibitive to obtain similar results to the next larger hexagonal array, which has 19 sites. On the other hand, it is slightly unreasonable to look for subdomains of a seven-site array, (7 < Nparticipating < 1). We will therefore take it that Figures 8 and 9 are approximations for what one will observe over a small region in a larger array. Such measurements are reported in, e.g., Figure 5 of the experimental work.25 We then argue that the different panels represent different regions of one large array, which is in the domainlocalized regime. The point to note is the amplitude of the signal. When the coupling is strong and very effective, bottom panel of Figure 9, the amplitude of the LDOS signal, is lowest. The amplitude significantly increases as the coupling becomes weaker. The origin of this change is very clear. The area under the signal is constant. As the bands merge, the signal spans a wider range in energy ()voltage) and hence its local value decreases. In terms of the LDOS it follows that eq 1.3 for the number of participating sites can be generalized to N

(N-1 Tr[Eˆ ii II((H ˆ - E)/∆E)]/ ∑ i)1

h ) ) 1/ Nparticipating(E

Tr[II((H ˆ - E)/∆E)])2 (7.3) There are two normalization conditions in (7.3). First is the factor N, which comes because of the definition (2.4), which N N implies that ∑i)1 〈Eˆ ii〉 ) ∑i)1 ni ) N. The other is to compensate for different intervals of given energy width ∆E having a different number of states.

Figure 8. STM spectrum computed using eq 5.3 for a seven site array including a Coulomb repulsion between electrons on the same site and without disorder in the site energies The three panels, for quite similar values of the lattice spacing, show the sharp closure of the band gap. Computed for an energy resolution of ∆E ) 0.01 eV. Variations in the site energies will smear this sharp change because they contribute to the broadening of the bands.

VIII. Concluding Remarks A coupling regime where charge is neither localized nor fully delocalized has been discussed. The effect can be seen either in the stationary states or as a restriction on the range of states that can be accessed by the migrating charge. The domain to which the charge is confined is that of neighboring sites, and the number of participating states is small compared to the total number of available sites. A domain-localized behavior is possible when adjacent units are not quite resonant so that the effective coupling ()exchange coupling/energy gap) between them is intermediate in strength. When the Coulomb blocking of charge migration is taken into account, one must distinguish two alternatives. One is when the variation in the site energies is limited in range and is smaller than the charging energy. In this case, at very weak coupling the states are strictly localized. It requires a finite site-site exchange coupling to close the Coulomb gap. Once this gap has been bridged then, upon a


Figure 9. An STM spectrum computed using eq 7.2 for a seven-site array including a Coulomb repulsion between electrons on the same site and with 5% variation in the site energies. The two panels, at quite different values of the lattice compression, contrast the spectrum in the localized and the delocalized regimes. Note the change in the scale between the two panels and the intermediate values of the amplitude of the spectrum shown in Figure 8. In the text we argue that this variation can be used to probe for a domain-localized regime.

further increase in the site-site coupling, domain localization sets in. A further increase brings about a complete delocalization. The second case is when the charging energy is comparable or lower than the variation in the site energies. In this case, domain localization is possible also for weaker site-site coupling. Acknowledgment. We thank R. Weinkauf and E. W. Schlag for discussions about their peptide cation experiments and U. Banin, J. R. Heath, S. -H. Kim, O. Millo, and R. S. Williams for discussions of STM. This work was supported by the von Humboldt Foundation and used computational facilities provided by SFB 377. F.R. thanks the “Action de Recherche Concerte´e” Lie`ge University, Belgium. References and Notes (1) Speiser, S. Chem. ReV. 1996, 96, 1953. (2) Depaemelaere, S.; Viaene, L.; Vanderauweraer, M.; Deschryver, F. C.; Hermant, R. M.; Verhoeven, J. W. Chem. Phys. Lett. 1993, 215, 649. (3) Jortner, J.; Bixon, M.; Wegewijs, B.; Verhoeven, J. W.; Rettschnick, R. P. H. Chem. Phys. Lett. 1993, 205, 451. (4) Jortner, J.; Bixon, M.; Langenbacher, T.; Michel-Beyerle, M. E. Proc. Natl. Acad. Sci. U.S.A. 1998, 95. (5) McConnell, H. M. J. Chem. Phys. 1961, 35, 508.

Remacle and Levine (6) Schatz, G. C.; Ratner, M. A. Quantum Mechanics in Chemistry; Prentice Hall: New York, 1993. (7) Weinkauf, R.; Schanen, P.; Metsala, A.; Schlag, E. W.; Buergle, M.; Kessler, H. J. Phys. Chem. 1996, 100, 18567. (8) Weinkauf, R.; Schanen, P.; Yang, D.; Soukara, S.; Schlag, E. W. J. Phys. Chem. 1995, 99, 11255. (9) Alivisatos, A. P. Science 1996, 271, 933. (10) Bawendi, M. G.; Steigerwald, M. L.; Brus, L. E. Annu. ReV. Phys. Chem. 1990, 41, 477. (11) Collier, C. P.; Vossmeyer, T.; Heath, J. R. Annu. ReV. Phys. Chem. 1998, 49, 371. (12) Markovich, G.; Collier, C. P.; Henrichs, S. E.; Remacle, F.; Levine, R. D.; Heath, J. R. Acc. Chem. Res. 1999, 32, 415. (13) Cave, R. J.; Newton, M. D. J. Chem. Phys. 1997, 106, 9213. (14) Collier, C. P.; Saykally, R. J.; Shiang, J. J.; Henrichs, S. E.; Heath, J. R. Science 1997, 277, 1978. (15) Baranov, L. Y.; Schlag, E. W. Z. Naturforsch., Sect. A 1999, 54, 387. (16) Toutounji, M. M.; Ratner, M. A. J. Phys. Chem. A 2000, 104, 8566. (17) Kiely, C. J.; Fink, J.; Brust, M.; Bethell, D.; Schiffrin, D. J. Nature 1998, 396, 444. (18) Kiely, C. J.; Fink, J.; Zheng, J. G.; Brust, M.; Bethell, D.; Schiffrin, D. J. AdV. Mater. 2000, 12, 640. (19) Chernyak, V.; Mukamel, S. J. Chem. Phys. 1996, 104, 444. (20) Takashi, A.; Mukamel, S. J. Chem. Phys. 1993, 100, 2366. (21) Heath, J. R.; Knobler, C. M.; Leff, D. V. J. Phys. Chem. B 1997, 101, 189. (22) Remacle, F.; Collier, C. P.; Heath, J. R.; Levine, R. D. Chem. Phys. Lett. 1998, 291, 453. (23) Henrichs, S.; Collier, C. P.; Saykally, R. J.; Shen, Y. R.; Heath, J. R. J. Am. Chem. Soc. 2000, 122, 4077. (24) Remacle, F.; Levine, R. D. J. Am. Chem. Soc. 2000, 122, 4084. (25) Kim, S.-H.; Meideiros-Ribeiro, G.; Ohlberg, D. A. A.; Williams, R. S.; Heath, J. R. J. Phys. Chem. B 1999, 103, 10341. (26) Remacle, F.; Levine, R. D. J. Phys. Chem. A 2000, 104, 10435. (27) Remacle, F.; Levine, R. D. P. Natl. Acad. Sci. U.S.A. 2000, 97, 553. (28) Levine, R. D. AdV. Chem. Phys. 1987, 70, 53. (29) Levine, R. D. J. Stat. Phys. 1988, 52, 1203. (30) Remacle, F.; Levine, R. D. J. Chem. Phys. 1993, 99, 2383. (31) Heller, E. J. Phys. ReV. A 1987, 35, 1360. (32) Coulson, C. A.; Fischer, I. Philos. Mag. 1949, 40, 386. (33) Remacle, F.; Levine, R. D. J. Chem. Phys. 2000, 113, 4515. (34) Slater, J. C. Quantum Theory of Molecules and Solids; McGrawHill: New York, 1963; Vol. I. (35) Parr, R. G. Quantum Theory of Molecular Electronic Structure; Benjamin: New York, 1963. (36) The Unitary Group for the EValuation of Electronic Energy Matrix Elements; Hinze, J., Ed.; Springer: Berlin, 1981; Vol. 22. (37) Paldus, J.; Cizek, J.; Hubac, I. Int. J. Quantum Chem. 1974, S8, 293. (38) Hubbard, J. Proc. R. Soc. 1963, 276, 238. (39) Pariser, R. J. Chem. Phys. 1956, 24, 250. (40) Free Electron Theory of Conjugated Molecules; Platt, J. R., Ed.; Wiley: New York, 1964. (41) Kemble, E. C. The Fundamental Principles of Quantum Mechanics; Dover: New York, 1958. (42) Mirsky, L. An Introduction to Linear Algebra; Dover: New York, 1990. (43) Remacle, F.; Levine, R. D. J. Chem. Phys. 1999, 110, 5089. (44) Mott, N. F. Metal-Insulator Transitions; Taylor & Francis: London, 1990. (45) Zallen, R. The Physics of Amorphous Solids; Wiley: New York, 1983. (46) Medeiros-Ribeiro, C.; Ohlberg, D. A. A.; Williams, R. S.; Heath, J. R. Phys. ReV. B 1999, 59, 1633. (47) Bracewell, R. N. The Fourier Transform and its Applications; McGraw-Hill: Singapore, 1986. (48) Banin, U.; Cao, Y.; Katz, D.; Millo, O. Nature 1999, 400, 542. (49) Chen, C. J. Unified Perturbation Theory for STM and SFM. In Scanning Tunneling Spectroscopy III; Wiesendanger, R., Gu¨ntherodt, H.J., Eds.; Springer-Verlag: Berlin, 1993; Vol. 29, p 141. (50) Lang, N. D. STM Imaging of Single-Atom Adsorbates on Metal. In Scanning Tunneling Spectroscopy III; Wiesendanger, R., Gu¨ntherodt, H.-J., Eds.; Springer-Verlag: Berlin, 1993; Vol. 29, p 7.

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