Effectively Unpaired Electrons for Singlet States: From Diatomics to Graphene Nanoclusters

Abstract

Formal and computational models within the effectively unpaired electron (EUE) theory are reviewed and extended. In the first part, we analyze open-ended aspects of the existing EUE measures and find additional advantages of the Head-Gordon index (2003) over the very first (Yamaguchi et al. 1978) index. In particular, for ground states the Head-Gordon index estimates an average occupation of virtual holes and particles, which occur due to electron correlation. Additional hole-particle indices for describing EUE are proposed and analyzed. The second part of the paper is focuses on practical aspects and EUE computational schemes in small molecules (at the ab initio level) and large-scale polyaromatic and graphene-like structures (at the semi-empirical level). Here the unrestricted Hartree-Fock (UHF) schemes and their recently proposed simplistic versions turn out to be a suitable tool producing meaningful EUE characteristics for the extended π-electron systems (with number of carbon atoms ~10³ and more) in a fast and simple way. We emphasize that UHF solutions should be regarded not as invalid spin-contaminated states but as precursors of the appropriate spin-projected states of the Lowdin’s extended Hartree-Fock type. The influence of the static and variable electric fields on π-electron systems is also studied. It is shown that strong perturbations drastically increase the electron unpairing in aromatic hydrocarbons, especially those with the initially stable Clar-type structure.

Dedicated to Late Prof. O.V. Shishkin.

Appendices

Appendix A: Duality Symmetry and Generalized EUE Indices

In this Appendix we clarify the cause for postulating symmetry relation (6.18). For this aim we introduce a formal operation which can be named the duality transformation and which is well known in multilinear algebra as the Hodge star operation, or Hodge dual [132]. In the RDM theory an equivalent transformation was applied in [19, 133], without recognizing it as a Hodge dual. The following simple example helps to explain this notion in the more familiar terms of many-electron state vectors.

We consider a two-electron problem in the basis of five spin-orbitals

$$ \left\{ {\left| {\chi_{1} } \right\rangle ,\left| {\chi_{2} } \right\rangle ,\left| {\chi_{3} } \right\rangle \left| {\chi_{4} } \right\rangle ,\left| {\chi_{5} } \right\rangle } \right\}. $$

(A1)

Let the ket

$$ \left| {\Psi _{{^{ [ 2 ]} }} } \right\rangle = \left| {\chi_{1} \chi_{2} } \right\rangle $$

(A2)

be the two-electron Slater determinant built from $\left| {\chi_{1} } \right\rangle$ and $\left| {\chi_{2} } \right\rangle$. By definition, the dual ket, $\left| {\Psi _{[3]}^{ *} } \right\rangle$, is built up from the rest spin-orbitals, giving the three-electron determinant:

$$ \left| {\Psi _{[3]}^{ *} } \right\rangle = \left| {\chi_{3} \chi_{4} \chi_{5} } \right\rangle . $$

(A3)

In the same basis (A1), the maximal Slater determinant $\left| {\Psi _{ \hbox{max} } } \right\rangle$ is

$$ \left| {\Psi _{ \hbox{max} } } \right\rangle = \left| {\chi_{1} \chi_{2} \chi_{3} \chi_{4} \chi_{5} } \right\rangle . $$

(A4)

It is not difficult understand that we can produce $\left| {\Psi _{[3]}^{ *} } \right\rangle$ from $\left| {\Psi _{ \hbox{max} } } \right\rangle$ by annihilating in Eq. (A4) the state vector (A2). More exactly, apart from a prefactor we have

$$ \left| {\Psi _{[3]}^{ *} } \right\rangle = \left\langle {{\Psi _{{^{ [ 2 ]} }} }} \mathrel{\left | {\vphantom {{\Psi _{{^{ [ 2 ]} }} } {\Psi _{ \hbox{max} } }}} \right. \kern-0pt} {{\Psi _{ \hbox{max} } }} \right\rangle . $$

(A5)

They say that the obtained three electron state $\left| {\Psi _{[3]}^{ *} } \right\rangle$ is the Hodge dual of the two-electron state $\left| {\Psi _{{^{ [ 2 ]} }} } \right\rangle$.

Now consider the respective 1-RDMs. In notation of Sect. 6.6, we have from determinants (A2) and (A3) the usual $D_{1}^{\text{so}}$ matrices in the form of projectors on occupied spin-orbitals of the respective determinants:

$$ D_{1}^{\text{so}} (\left| {\Psi _{{^{ [ 2 ]} }} } \right\rangle ) = \left| {\chi_{1} } \right\rangle \left\langle {\chi_{1} } \right| + \left| {\chi_{2} } \right\rangle \left\langle {\chi_{2} } \right|, $$

$$ D_{1}^{\text{so}} (\Psi _{[3]}^{ *} ) = \left| {\chi_{3} } \right\rangle \left\langle {\chi_{3} } \right| + \left| {\chi_{4} } \right\rangle \left\langle {\chi_{4} } \right| + \left| {\chi_{5} } \right\rangle \left\langle {\chi_{5} } \right|. $$

We see that

$$ D_{1}^{\text{so}} (\Psi _{[3]}^{ *} ) = I - D_{1}^{\text{so}} (\Psi ^{ [ 2 ]} ) $$

(A6)

where unity operator I is a projector on all five spin-orbitals from Eq. (A1).

This line of reasoning can be directly extended to a general case including exact state vectors as well. It the general case we start with a r-dimensional spin-orbital basis $\{ \left| {\chi_{k} } \right\rangle \}_{1 \le k \le r}$ and build the respective maximal determinant $\left| {\Psi _{ \hbox{max} } } \right\rangle = \left| {\chi_{1} \ldots \chi_{r} } \right\rangle$ (clearly, the only r-electron state vector is $\left| {\Psi _{ \hbox{max} } } \right\rangle \equiv \left| {\Psi _{ [r ]} } \right\rangle \,$). The given exact (or approximate) state-vector $\Psi _{ [N ]}$ produces the Hodge dual, as previously:

$$ \left| {\Psi _{ [r - N ]}^{ *} } \right\rangle = \left\langle {{\Psi _{ [N ]} }} \mathrel{\left | {\vphantom {{\Psi _{ [N ]} } {\Psi _{ \hbox{max} } }}} \right. \kern-0pt} {{\Psi _{ \hbox{max} } }} \right\rangle . $$

(A7)

Accordingly, relation (A6) is generalized to be

$$ D_{1}^{\text{so}} (\Psi _{ [r - N ]}^{ *} ) = I - D_{1}^{\text{so}} (\Psi _{ [N ]} ). $$

(A8)

This is the duality transformation in terms of 1-RDM. The analogous relation for $D_{2}^{\text{so}} (\Psi _{ [r - N ]}^{ *} )$ is somewhat more involved [133, 134]. The remarkable property of the Hodge duality transformation is its ability to preserve correlation operator $\varDelta_{2}^{\text{so}}$ in Eq. (6.45), as it is first shown in [19]. The related expression is given in [135]. Thus, the other correlation matrices, e.g., $D^{\text{eff}}$, must be the same as well. It is worth mentioning in passing that in [128] and many subsequent papers, a somewhat inconvenient terminology is used for RDMs $D_{{}}^{\text{so}} (\Psi _{ [r - N ]}^{ *} )$—the latter are loosely identified with hole RDMs. Certainly, it leads to confusion and even misinterpretation, since generally such RDMs have no relation to the genuine, ‘physical’, hole RDMs discussed in Sect. 6.4 and in [16]. We prefer to refer to them as the dual RDMs [16].

We now have to sum over spin indices, making spin trace in Eq. (A8). As a result, the dual charge density matrix is yielded, viz.

$$ D(\Psi _{ [r - N ]}^{ *} ) = 2 - D(\Psi _{ [N ]} ). $$

(A9)

Then the NOON spectrum of the dual charge density matrix is simply a set $\{ 2 - \lambda_{k} \}$ where we imply that the initial NOON spectrum is the set $\{ \lambda_{k} \}$. Recalling that EUE characteristics of the dual state (A7) should be the same as in the initial state $\Psi _{ [N ]}$, the identity

$$ D^{\text{eff}} (\Psi _{ [N ]} ) = D^{\text{eff}} (\Psi _{ [r - N ]}^{ *} ) $$

(A10)

is necessitated. Taking into account Eqs. (6.1), (6.5) and (6.5′) we arrive at the relation

$$ \sum\limits_{k} {\,f(\lambda_{k} )\left| {\varphi_{k} } \right\rangle \left\langle {\varphi_{k} } \right|} = \sum\limits_{k} {\,f(2 - \lambda_{k} )\left| {\varphi_{k} } \right\rangle \left\langle {\varphi_{k} } \right|} , $$

(A11)

from whence Eq. (6.18) immediately follows, that is

$$ f(\lambda ) = f(2 - \lambda ). $$

(A12)

The requirement (A12) allows us to specify a general dependence $\lambda_{{}}^{\text{eff}} = f(\lambda_{{}} )$, namely, $\lambda_{{}}^{\text{eff}}$ is a nonnegative definite function of argument $|\lambda - 1|$, with boundary values $f(0) = f(2) = 0.$ Eqs. (6.7′) and (6.14) are evidently of this type. Rather general types of the functions can be proposed as ‘q-extensions’ of Eqs. (6.9) and (6.15). These are

$$ N_{\text{odd}}^{{}} \,[q] = \sum\limits_{k} {\,(1 - |\lambda_{k} - 1|^{2} )^{q} } , $$

(A13)

$$ N_{{{\text{eff}}\,}}^{{}} \,[q] = \sum\limits_{k} {\,(1 - |\lambda_{k} - 1|)^{q} } , $$

(A14)

where $q \ge 1$. We see that $N_{\text{odd}}^{{}} \,[1]$ and $N_{{{\text{eff}}\,}}^{{}} \,[1]$ produce the usual $N_{\text{odd}}^{{}}$ and $N_{\text{eff}}^{{}}$ measures, respectively. The choice $q = 2$ in Eq. (A13) leads to

$$ N_{\text{odd}} [2] = \sum\limits_{k} {\,[1 - (\lambda_{k} - 1)^{2} )]} , $$

(A15)

which is the modified Head-Gordon index from [5]. This expression is trivially equivalent to Eq. (6.94).

Appendix B: Density Matrix and NOON for QCTB

We consider here in more detail the QCTB model described in Sect. 6.13. Having at hand the effective Hamiltonian matrices (6.91), we straightforwardly derive projector matrices $\rho_{\alpha }$ and $\rho_{\beta }$ by using the well known expressions connecting Hamiltonians and respective projectors [19, 136, 137]). Let h be the Hermitian operator, such that exactly n eigenvalues of h lie below zero, and P be the projector on the corresponding eigenvectors. Then

$$ P = (I - h/|h|)/2, $$

(B1)

where $|h| = \,\,\,[(h)^{2} ]^{1/2}$ is the modulus of operator h. Further, let one-electron Hamiltonian matrix $h^{\delta }$ be defined as follows:

$$ h^{[\delta ]} = - \left( {\begin{array}{*{20}c} {\delta \,I} & B \\ {B^{ + } } & {\,\, - \delta \,I} \\ \end{array} } \right). $$

In particular, $h^{\alpha } = h^{[\delta ]}$, $h^{\beta } = h^{[ - \delta ]}$. Then, by applying Eq. (B1) to $h = h^{[\delta ]}$, we obtain the corresponding projector

$$ P^{[\delta ]} = \frac{1}{2}\left( {\begin{array}{*{20}c} {I + \delta (\delta^{2} I + BB^{ + } )^{ - 1/2} } & {B(\delta^{2} I + B^{ + } B)^{ - 1/2} } \\ {(\delta^{2} I + B^{ + } B)^{ - 1/2} B^{ + } } & {I - \delta (\delta^{2} I + B^{ + } B)^{ - 1/2} } \\ \end{array} } \right). $$

(B2)

In derivation, the block-diagonal structure of $(h^{[\delta ]} )^{2}$ is used, that is

$$ (h^{[\delta ]} )^{2} = \left( {\begin{array}{*{20}c} {\delta^{2} I + BB^{ + } } & 0 \\ 0 & {\delta^{2} I + B^{ + } B} \\ \end{array} } \right). $$

Equation (B2) was earlier derived by another technique for the special closed π-shells with alternating electronegativity [138]. Obviously, setting $\delta = 0$, we return to the Hall formula (6.90). By recalling Eq. (6.91) we have

$$ \rho_{\alpha } = P^{[\delta ]} ,\;\;\rho_{\beta } = P^{[ - \delta ]} . $$

(B3)

Putting together Eqs. (B2) and (B3), we get from Eq. (6.10) the main result:

$$ D = \left( {\begin{array}{*{20}c} I & {B(\delta^{2} I + B^{ + } B)^{ - 1/2} } \\ {(\delta^{2} I + B^{ + } B)^{ - 1/2} B^{ + } } & I \\ \end{array} } \right). $$

(B4)

The problem of diagonalizing this $D$ is a quite elementary, and the full NOON spectrum takes the form

$$ \lambda_{i}^{{}} = 1 + \varepsilon_{i} /\sqrt {\delta^{2} + \varepsilon_{i}^{2} } ,\;\,\lambda_{a}^{{}} = 1 - \varepsilon_{a} /\sqrt {\delta^{2} + \varepsilon_{a}^{2} } , $$

(B5)

where $1 \le i,\,a \le n$, and nonnegative quantities $\varepsilon_{i} \equiv \left| {\varepsilon_{i} } \right|$, as well as $\varepsilon_{a} \equiv \left| {\varepsilon_{a} } \right|$, are eigenvalues of $(B^{ + } B)^{1/2}$, that is $\{ \varepsilon_{i} \}$ is the bipartite graph spectrum. From Eq. (B5) the main EUE indices within QCTB are easily deduced. For instance,

$$ N_{\text{odd}}^{{}} = 2\delta^{2} \sum\limits_{i = 1}^{n} {(\delta^{2} + \varepsilon_{i}^{2} )^{ - 1} } . $$

(B6)

Remark also an evident symmetry of the corresponding hole and particle occupancies, defined by Eq. (6.41′):

$$ \{ 1 - \varepsilon_{i} /\sqrt {\delta^{2} + \varepsilon_{i}^{2} } \}_{{}} = \{ 1 - \varepsilon_{a} /\sqrt {\delta^{2} + \varepsilon_{a}^{2} } \} . $$

(B7)

that follows from Eq. (B5). In other words, the hole and particle occupancy spectra are identical for this π-model.

As a matter of fact, the hole and particle occupancies are identical for any bipartite networks treated within π-approximation, up to FCI/PPP. This is a simple corollary of the generalized pairing theorem of McLachlan [94] stating that the π-electron charge density matrix of the alternant hydrocarbons is of the form

$$ D = \left( {\begin{array}{*{20}c} I & \partial \\ {\partial^{ + } } & I \\ \end{array} } \right), $$

(B8)

where the $2p_{z}$ AO basis set is ordered as in Eq. (6.89), and $\partial$ defines the inter-sublattice bond order matrix. Clearly, the corresponding NOON spectrum $\{ \lambda_{k} \}$ is

$$ \{ 1 + \sqrt {\mu_{i} } \} ,\,\,\{ 1 - \sqrt {\mu_{a} } \} $$

(B9)

where $\mu_{i}$ (or $\mu_{a}$) are eigenvalues of $\partial^{ + } \partial$, and $1 \le i,\,a \le n$. As a result, the initial π-NOON spectrum is symmetrical in respect to the point $\lambda = 1$. From Eq. (B9) we deduce that indeed the respective hole and particle π-occupancies, defined as in Eq. (6.41′), are identically the same:

$$ \{ 1 - \sqrt {\mu_{i} } \} = \{ 1 - \sqrt {\mu_{a} } \} . $$

(B10)

Interestingly, an initio data [9, 11] approximately follow Eqs. (B9) and (B10). Incidentally, it follows, from this discussion, that the hole occupancy distribution $\{ 1 - \sqrt {\mu_{i} } \}$ (generally $\{ 2 - \lambda_{i} \}_{1 \le i \le n}$) is sufficient for considering EUE problems. For instance, instead of plotting NOON spectrum $\{ \lambda_{k} \}$, one can plot only hole occupancy spectrum $\{ 2 - \lambda_{i} \}$ as even more suitable in the EUE context. This occupancy spectrum is in fact the second half of the typical π-NOON spectra which were presented in Tables 6.6, 6.7, and Fig. 6.4.

Appendix C: Generalized Hole-Particle Indices

Here we analyze the main EUE indices in terms of hole-particle quantities. We begin with the representation

$$ D = 2\rho + \varDelta D, $$

(C1)

where $\rho$ is of the form (6.36), and $\left| {\varphi_{i} } \right\rangle$ are the natural orbitals of the state in question, so $\varDelta D$ commutes with $\rho$. Then, using the same notation, as in Eq. (6.41), we obtain the spectral resolution

$$ \varDelta D = - \sum\limits_{i \le n} {\varDelta_{i} \,\left| {\varphi_{i} } \right\rangle \left\langle {\varphi_{i} } \right|} + \sum\limits_{a > n} {\,\lambda_{a}^{{}} \left| {\varphi_{a} } \right\rangle \left\langle {\varphi_{a} } \right|} , $$

(C2)

where

$$ \varDelta_{i} \equiv 2 - \lambda_{i} $$

(C3)

are new nonnegative quantities ($0 \le \varDelta_{i} < 1$, and $i \le n$), and $\lambda_{a}^{{}}$ are related to ‘virtual’ natural orbitals. We see that correlation correction matrix $\varDelta D$ has a clear hole-particle structure: $\varDelta_{i} \,$ are the occupancy numbers for the holes, and $\lambda_{a}^{{}}$ are the same for the particles. In manipulations the identity

$$ \sum\nolimits_{i} {\varDelta_{i} } = \sum\nolimits_{a} {\lambda_{a} } $$

(C4)

will be useful as well. It follows from Eqs. (C1), (C2), and normalization (6.2).

Due to the diagonal form (C2) we trivially have the diagonal form of the matrix $|\varDelta D|\,$ defined by Eq. (6.96):

$$ |\varDelta D|\, = \sum\nolimits_{i} {\,\varDelta_{i} \,\,} \,\left| {\varphi_{i} } \right\rangle \left\langle {\varphi_{i} } \right|\, + \sum\nolimits_{a} {\,\,} \lambda_{a}^{{}} \left| {\varphi_{a} } \right\rangle \left\langle {\varphi_{a} } \right| . $$

(C5)

But this is the same as the hole-particle density in Eq. (6.41), that is

$$ D^{\text{h - p}} = |\varDelta D|. $$

(C6)

It is essential that under duality transformation (A9) the holes and particles in Eq. (C2) change place, so identity (A10) satisfies automatically for $D^{\text{eff}} = D^{\text{h - p}}$.

The appropriate q-extended ($q \ge 1$) hole-particle indices can be cast explicitly into the form

$$ N_{\text{h - p}}^{{}} [q] = {\text{Tr}}(|\varDelta D|\,^{q} ) = \sum\nolimits_{i} {\varDelta \,_{i}^{q} } + \sum\nolimits_{a} {\lambda_{a}^{q} } . $$

(C7)

Particularly,

$$ N_{\text{h - p}}^{{}} [2] = \sum\nolimits_{i} {\varDelta \,_{i}^{2} } + \sum\nolimits_{a} {\lambda_{a}^{2} } = ||\varDelta D\,||^{2} . $$

(C8)

The previously defined EUE indices can be rewritten in terms of the correlation-dependent quantities {$\varDelta_{i}$, $\lambda_{a}$}:

$$ N_{\text{odd}}^{{}} = \sum\limits_{1 \le i \le n} {(4\,\varDelta_{i} + \varDelta \,_{i}^{2} ) + \sum\limits_{a > n} {\,\lambda_{a}^{2} } } , $$

(C9)

$$ N_{\text{eff}}^{{}} = \sum\nolimits_{\,i\,} {\varDelta_{i} } + \sum\nolimits_{a} {\lambda_{a}^{{}} } = 2\sum\limits_{1 \le i \le n} {\,\varDelta_{i} } , $$

(C10)

$$ N_{\text{odd}}^{{}} \,[2] = \sum\limits_{1 \le i \le n} {[\varDelta_{i} (2 - \varDelta_{i} )]^{2} + \sum\limits_{a > n} {\,[\lambda_{a} (2 - \lambda_{a} } )]^{2} } . $$

(C11)

where we used identity (C4).

For slightly correlated systems, the most important are the first order terms in $\varDelta_{i}$ and $\lambda_{a}$. It gives $N_{\text{odd}}^{{}} \cong 4\sum {\varDelta_{i} }$, so

$$ N_{\text{odd}}^{{}} \cong 2N_{\text{eff}}^{{}} , $$

(C12)

and this goes back to the rude estimation, Eq. (6.80). It is interesting that the exact interrelation $2N_{\text{eff}}^{{}} - N_{\text{odd}}^{{}} = \,||\varDelta D\,||^{2}$ is true. Likewise, the first-order estimation of the modified Head-Gordon index (6.94), that is Eq. (C11), is null:

$$ N_{\text{odd}}^{{}} \,\,[2] \cong 0. $$

(C13)

Indeed, Eq. (C11) contains only the second-order and higher-order terms:

$$ N_{\text{odd}}^{{}} \,\,\,\,[2] \cong 4\,(\,\sum\nolimits_{i} {\,\varDelta \,_{i}^{2} } \, + \sum\nolimits_{a} {\lambda_{a}^{2} } ) = 4||\varDelta D\,||^{2} . $$

(C14)

The above simple analysis now elucidates how small contributions from $\varDelta_{i}$ and $\lambda_{a}$ are essentially suppressed in the $N_{\text{odd}}^{{}} \,\,\,[2]$ and $N_{\text{h - p}}^{{}} [2]$ indices. As a rule, these small contributions appear mainly from dynamical correlations. For instance, MP2 (the Moller-Plesset second-order perturbation theory) normally produce the contributions of this kind. Evidently, they have no direct relation to diradicality and polyradicality, and the $N_{\text{odd}}^{{}} \,\,\,[2]$ and $N_{\text{h - p}}^{{}} [2]$ indices should be rather small without a significant contribution from non-dynamical correlation. This is a good property of the generalized indices such as (6.94) and (C8), and apparently, this is the basic reason why $N_{\text{odd}}^{{}} \,\,\,[2]$ is systematically employed in papers [9, 11, 122, 124] for analyzing the unpaired electrons in large PAHs. At the same time, the dynamical correlation cannot fully ignored, and the problem of an optimal quantification remains.