Single-Molecule Conductance Theory Using Different Orbitals for Different Spins: Applications to π-Electrons in Graphene Molecules

Abstract

In the present paper, we consider many-body π-electron models and computational tools for treating single-molecule conductance in middle-size graphene molecules. Our study highlights the importance of accounting for long-range interactions and electron-correlation effects which are crucial for a correct description of electron transmission in conjugated systems. Here we construct one-electron Green’s function matrix for the half-projected Hartree-Fock method implementing the different orbitals for different spins (DODS) approach. Moreover, the simplified DODS in the form of quasi-correlated tight-binding (QCTB) and related models are invoked. We compare these electron-correlation models with the usual tight-binding (TB) approximation and show that TB is actually incorrect as a single-molecule conductance theory of graphenic and similar structures. In our specific applications, we calculate the conductance spectra for small graphene nanoflakes and find, in particular, that “zigzag” connections can afford significantly higher electron transmission than “armchair” connections.

Appendices

Appendices

1.1 Appendix A: Construction of HPHF Green’s Function

Before treating in detail GF for HPHF, it is sensible to consider a standard general expression of GF. Let us first rewrite Eq. (22.2) in an appropriate spectral form:

$$ R=\sum \limits_i\;\frac{\left|{\phi}_i\right\rangle \left\langle {\phi}_i\right|}{E+i{0}^{+}-{\varepsilon}_i}+\sum \limits_a\;\frac{\left|{\phi}_a\right\rangle \left\langle {\phi}_a\right|}{E+i{0}^{+}-{\varepsilon}_a},\vspace*{-3pt} $$

(22.A1)

where ε _i and ε _a are Koopmans orbital energies, that is eigenvalues of h; |ϕ _i〉 and |ϕ _a〉 are corresponding occupied and virtual MOs (eigenkets of h), respectively. In fact, the structure of Eq. (22.A1) remains valid in a more general setting (as in Eq. (5.76) from Ref. [1]). In doing so, |ϕ _i〉 and |ϕ _a〉 should be replaced by the so-called Dyson orbital $\left|{d}_i^{+}\right\rangle$ for electron detachment, and by Dyson orbital $\left|{d}_a^{-}\right\rangle$ for electron attachment; they may be nonorthogonal to each other and even be linear dependent [50]. In addition, ε _i and ε _a are replaced with transition energies $\Delta {E}_i^{+}$ and $\Delta {E}_a^{-}$, respectively. Explicitly, $\Delta {E}_i^{+}={E}^N-{E}_i^{N-1}$ (negative ionization potential), and $\Delta {E}_a^{-}={E}_a^{N+1}-{E}^N$ (electron affinity). It gives the most general (Lehmann type) spectral representation of GF for N-electron system:

$$ R=\sum \limits_i\;\frac{\left|{d}_i^{+}\right\rangle \left\langle {d}_i^{+}\right|}{E+i{0}^{+}-\Delta {E}_i^{+}}+\sum \limits_a\;\frac{\left|{d}_a^{-}\right\rangle \left\langle {d}_a^{-}\right|}{E+i{0}^{+}-\Delta {E}_a^{-}}.\vspace*{-3pt} $$

(22.A2)

Now we turn to the HPHF model for which the variational Koopmans-like orbitals were constructed in Ref. [28]. We will need the standard (Hermitian) matrix projectors onto the occupied spin-up and spin-down MOs, that is

$$ {\rho}^{\alpha }=\sum \limits_{i=1}^n\left|{\phi}_i^{\alpha}\right\rangle \left\langle {\phi}_i^{\alpha}\right|,\kern1em {\rho}^{\beta }=\sum \limits_{i=1}^n\left|{\phi}_i^{\beta}\right\rangle \left\langle {\phi}_i^{\beta}\right|, $$

(22.A3)

along with a non-Hermitian matrix projector U which is generated by overlapping of ρ ^α and ρ ^β:

$$ U={\rho}^{\alpha }{\left({\rho}^{\beta }{\rho}^{\alpha}\right)}^{-1}{\rho}^{\beta }.\vspace*{-3pt} $$

(22.A4)

Matrix inversion here should be understood as the Moore-Penrose pseudoinverse (see, e.g., Ref. [51]). The next are the Fockian matrices, f _α and f _β, associated with the above projectors:

$$ {f}_{\alpha }=h+J\left({\rho}^{\alpha }+{\rho}^{\beta}\right)-K\left({\rho}^{\alpha}\right),\kern1em {f}_{\beta }=h+J\left({\rho}^{\alpha }+{\rho}^{\beta}\right)-K\left({\rho}^{\beta}\right)\vspace*{-24pt} $$

(22.A5)

$$ {f}_U=h+J\left(U+{U}^{+}\right)-K(U), $$

(22.A6)

with J and K being, respectively, standard Coulomb and exchange (super)operators due to Roothaan. In above, h is a core Hamiltonian which includes not only h ⁰ but electron-nuclear attraction terms.

Then we can derive the HPHF variational equation for $\left|{d}_i^{+}\right\rangle$, based on Eqs. (35) and (36) from Ref. [28]. We first define the (nonnormalized) charge density matrix, D, at the HPHF level:

$$ D={\rho}^{\alpha }+{\rho}^{\beta }+\xi\;\left(\;U+{U}^{+}\right), $$

(22.A7)

where ξ is a pseudodeterminant of ρ ^α ρ ^β (i.e., the last nonnull (nth) coefficient of its characteristic polynomial). This D serves as an auxiliary matrix in the generalized eigenvalue problem of the form:

$$ \Theta {D}^{-1}\left|{d}_i^{+}\right\rangle =\left({E}_{\mathrm{HPHF}}-\Delta {E}_i^{+}\right)\left|{d}_i^{+}\right\rangle, $$

(22.A8)

where

$$ \Theta ={\rho}^{\alpha}\left({E}_{\rho }-{f}_{\alpha}\right){\rho}^{\alpha }+{\rho}^{\beta}\left({E}_{\rho }-{f}_{\beta}\right){\rho}^{\beta }+\xi\;\left[U\left({E}_U-{f}_U\right)U+\mathrm{h}.\mathrm{c}.\right], $$

(22.A9)

and E _ρand E _U are usual UHF-like energies for projectors ρ ^α, ρ ^β and U, U ⁺, respectively. Moreover, E _HPHF (i.e., E _N needed for $\Delta {E}_i^{+}$) is known beforehand: E _HPHF = (E _ρ + ξ E _U)/(1 + ξ). The eigenvalue problem for $\left|{d}_a^{-}\right\rangle$and $\Delta {E}_a^{-}$ is formulated likewise. Namely, the relevant eigenvalue problem for $\Delta {E}_a^{-}$ can be obtained from Eqs. (22.A5), (22.A6), (22.A7), (22.A8), and (22.A9) by replacing all projectors by their “vacant” counterparts (ρ ^α → I − ρ ^α,U → I − U etc.), but leaving all the Fockians, Eqs. (22.A5) and (22.A6), unchanged. At last, in order to get the resulting R ^HPHF from the eigensolutions of Eq. (22.A8) and their counterparts for $\Delta {E}_a^{-}$, we directly apply Eq. (22.A2).

We now shortly discuss the selection rules for matrix R ₀, i.e., for GF matrix elements at E = E _F, neglecting energy broadening effects. The main rule is that for any correct bipartite-symmetry description we have the same block skew-diagonal structure of R ₀ as in the underlying TB Hamiltonian, Eq. (22.5). Thus,

$$ {R}_0=\kern0.36em \left(\begin{array}{l}\;0\kern0.72em {R}_{\ast \circ}\\ {}\;{R}_{\circ \ast}\kern0.36em 0\end{array}\right). $$

(22.A10)

This equation for TB is trivial because ${R}_0^{\mathrm{TB}}=-{\left({h}^0\right)}^{-1}$. Eq. (22.A10) is indeed the selection rule since it states that there are no nonzero elements of GF for (a,b) connections with a and b belonging simultaneously to the same atomic set, either the starred or unstarred set. Far less trivial is the fact that Eq. (22.A10) holds true for GF at the π-FCI level, as was stated rigorously in the important theorem obtained in Ref. [18]. Therefore, Eq. (22.A10) as originating from the bipartite symmetry, should be valid for any consistent π-approximation not violating a topological symmetry. The same selection is exactly fulfilled for QCTB [20], and it is not so difficult to prove the same rule at the HPHF level as well.

1.2 Appendix B: Approximate Versus “Exact” π-Electron Results for Small Aromatics

In order to estimate reliability of the results obtained by various approximate π-models, we consider briefly the formally exact π-electron theory based on the well-known FCI method (e.g., see [52]). In our computations, we will follow the previously proposed FCI matrix algorithm; for additional references see Ref. [53] where a suitable FCI approach to calculating Dyson orbitals is given. As to the MSE problems, the first important results at the π-FCI level were given only recently in Ref. [18]. In what follows, the FCI results we present here will be taken as the benchmark data against which all the others must be compared.

One special point concerns the actual Fermi energy E _F that should be used to ensure Eq. (22.A10) for bipartites. In Ref. [18] the E _F value is not given explicitly. At the same time, for bipartites the remarkable Hush and Pople theorem is valid at the π-electron Hartree-Fock level [54], as well as at the FCI level [55]. From this theorem it follows that E _F = W _C + γ _C/2, where W _C is the standard effective ionization potential, and γ _C is the π-electron one-center Coulomb repulsion integral for the carbon atom. Just this choice of E _F ensures the validity of Eq. (22.A10) and other properties of GF for bipartites.

In our specific π-electron computations, we use standard π-electron parameters (in eV): resonance integral of the aromatic π-bond β ₀ =  − 2.4 ; W _C = 0, γ _C = 11.13, and two-center repulsion integrals due to Ohno. For QCTB computations, we adopt δ = 7/24 and E _F = 0. The idealized regular geometry was taken for the carbon backbone in all studies of conjugated π-structures (1.4 Å for CC bond lengths, etc.).

Now, let us say few words about the supplementary rescaling of the GF matrix elements for RHF, HPHF, and FCI, following the procedure from Ref. [20]. This was proposed in order to avoid an inevitably large gap between different approaches. When multiplying RHF, HPHF, and FCI matrix elements of GF by the scaling factor β ₀/(β ₀ − γ ₁₂/2) we make them comparable with their TB and QCTB counterparts. In particular, in the ethylene molecule the respective (1,2) elements, ${\left({R}_0^{\mathrm{TB}}\right)}_{1,2}$ and ${\left({R}_0^{\mathrm{RHF}}\right)}_{1,2}$ for the CC π bond, become identical and equal to 1.

Now we describe the results of comparison between π-FCI (the most rigorous π-approach) and main approximations (HPHF, RHF, QCTB, and TB). In addition, we tentatively and preliminary propose an improvement of QCTB in order to include long-range interactions not incorporated in the topological schemes. We simply do the first iteration of an usual self-consistent RHF procedure based on the TB (Hückel) density matrix as a start. It gives us a modified one-electron Hamiltonian of the correct block structure as in Eq. (22.5) for h ⁰. Then, expressions of the same type as in Eqs. (22.7), (22.8), (22.9), and (22.10) are applied in order to compute an approximated GF. This approach will be termed the quasi-correlated long-range interaction (QCLRI) model, and the respective GF will be denoted by G ^QCLRI. More detail will be given elsewhere.

Let us examine the numerical results presented in Table 22.3. The specific connections (^∗,∘) are shown in Table 22.3 by stars and cycles. We see that HPHF provides the best (in respect to FCI) results whereas there are marked quantitative deviations of QCTB from FCI. Especially large deviations from FCI occur for TB. It is worth paying attention to a good quality of the RHF results for the considered small aromatic molecules. In fact, RHF provides here better results than TB and even QCTB. However, RHF calls for much more computational efforts, but more essential is that RHF is not appropriate for computing GF in extended π-systems (see Sect. 22.5). It is important for future applications to observe that QCLRI, i.e., the above-proposed simple π-scheme, surprisingly works almost as well as HPHF, at least for the considered molecules. It is noteworthy that, unlike QCTB, the QCLRI method possesses the size-consistency discussed in the last paragraph of Sect. 22.4).

Table 22.3 GF matrix elements R ₀ (E = E _F) for small aromatic molecules at the various levels of the theory

It is pertinent to understand now how significant in practice can be errors caused by lacking size-consistency in HPHF. A direct way to estimate actual inaccuracy due to the size inconsistency is to compute GF matrix elements in non-covalent intermolecular dimers of the chosen systems. Indeed, GF should be an additive-type size-consistent quantity (as closely related to the one-electron density matrix), and the same follows also from definition (22.A2). It means that the GF matrix of any noncovalent intermolecular dimer or complex, say, complex AB, must take the form of a direct sum when an average intermolecular distance goes to infinity. For example, in a dissociated dimer AB we have at the FCI level, R ^FCI[AB] = R ^FCI[A] ⊕ R ^FCI[B], and likewise for other size-consistent models, such as RHF, QCLRI, QCTB, and TB. Unfortunately, this is not the case of HPHF and related spin-projected Hartree-Fock models.

Let us examine the selected GF elements of the dimerized systems for the molecules studied in Table 22.3. For each dissociated dimer, its constituent monomeric parts A and B were situated at the intermolecular distance equal to 100 Å. Of course, FCI, QCLRI, RHF, QCTB, and TB obey the size-consistent requirement, so that the corresponding GF matrix elements in the initial monomer molecule and in the related parts of the dimer are exactly the same, and we do not repeat these data. At the same time, in the case of HPHF we obtain slightly different results for the monomer and the respective dimer subunits. We find the following HPHF values for GF elements under dissociation of the benzene, butalene, naphthalene, diphenylene, and naphtha[b]cyclobutadiene dimers:

$$ -0.475,-0.407,0.255,-0.142,-0.157 $$

These values should be compared with the respective values in the third column of Table 22.3. We see that in the dissociated dimers the deviation of GF elements from the ones obtained for the monomer are around of order 5%.