Local descriptors of dynamic and nondynamic correlation

TL;DR

This paper introduces real-space local descriptors for dynamic and nondynamic electron correlation, enabling orbital decomposition and quantification of correlation effects in various chemical systems.

Contribution

It presents novel local descriptors that distinguish dynamic and nondynamic correlation regions, applicable in multiple quantum chemistry methods.

Findings

Descriptors successfully differentiate correlation types in diverse systems

Global correlation measures correlate with chemical properties

Method integrates easily with existing computational frameworks

Abstract

Quantitatively accurate electronic structure calculations rely on the proper description of electron correlation. A judicious choice of the approximate quantum chemistry method depends upon the importance of dynamic and nondynamic correlation, which is usually assessed by scalar measures. Existing measures of electron correlation do not consider separately the regions of the Cartesian space where dynamic or nondynamic correlation are most important. We introduce real-space descriptors of dynamic and nondynamic electron correlation that admit orbital decomposition. Integration of the local descriptors yields global numbers that can be used to quantify dynamic and nondynamic correlation. Illustrative examples over different chemical systems with varying electron correlation regimes are used to demonstrate the capabilities of the local descriptors. Since the expressions only require orbitals and occupation numbers, they can be readily applied in the context of local correlation methods, hybrid methods, density matrix functional theory and fractional-occupancy density functional theory.

Equations

$$I_T$$

$$I_D$$

$$I_{\text{ND}}$$

$$I_T(\mathbf{r})$$

$$I_D(\mathbf{r})$$

$$I_{\text{ND}}(\mathbf{r})$$

Full text

Euskal Herriko Unibertsitatea] Kimika Fakultatea, Euskal Herriko Unibertsitatea, UPV/EHU, and Donostia International Physics Center (DIPC). P.K. 1072, 20080 Donostia, Euskadi, Spain \alsoaffiliationCurrently at Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley

Euskal Herriko Unibertsitatea] Kimika Fakultatea, Euskal Herriko Unibertsitatea, UPV/EHU, and Donostia International Physics Center (DIPC). P.K. 1072, 20080 Donostia, Euskadi, Spain \alsoaffiliation[Ikerbasque, Basque Foundation for Science] IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.

1 Introduction

A proper treatment of electron correlation is essential for an accurate description of a quantum chemical system. Electron correlation affects not only the electronic energy 1, 2, 3, 4 but also many molecular properties 5 such as chemical shifts, 6 the spin, 7 optical properties, 8 the bonding character, 9, 10, 11, 12 or three-center interactions. 13, 14, rodriguez:17pccp The term electron correlation was introduced in quantum chemistry in 1934 by Wigner and Seitz, 15 when they studied the cohesive energy of metals but it was Löwdin who defined the electron correlation energy as the exact energy minus the Hartree-Fock (HF) energy. 16 Löwdin’s expression measures an electronic energy difference that reflects the (electron correlation) effects missing in the single-determinant HF wavefunction. Since then, many different terms related to the electron correlation have been coined in order to classify and recognize the deficiencies of approximate computational methods. Hence, words such as dynamic, static, short- and long-range correlation are now commonly employed by computational and theoretical chemists. 17, 18, 19, 20, 21, 22

A reliable description of electron correlation effects in molecular systems involves the use of methods that present an unfavorable scaling with the system size. As a fitting solution, orbital localization schemes have been widely used to bring down the computational cost of wavefunction methods. The use of atom-centered expansions of many-body wavefunctions that benefit from the local character of atomic orbitals has also become a major topic of research over the past years. In either case, functions of localized nature are used to reduce the computational cost for large molecules. 23, 24, 25 Local descriptors of electron correlation can be likewise used to recognize the most important interactions and regions in a molecule, leading to cost-efficient ways of treating electron correlation. For instance, local descriptors of electron correlation can be used to screen the working orbital set or to generate a new set of orbitals including the most important correlation effects. If these local descriptors provide a separation of electron correlation effects, they could be also employed to combine different computational methods in the framework of hybrid methods. 26, 27, 28, 29

Dynamic and nondynamic correlation (DC and NDC hereafter) stand out among the terms used to classify electron correlation effects. 30 DC arises from the inability of HF wavefunction to model interelectronic cusps and dispersion interactions, whereas NDC stems from near-degeneracy of HF occupied and virtual orbitals. 20 NDC is essentially a system-dependent correlation effect that calls for a specific treatment, while DC shows a universal character. 30, 31, 32 Many methods provide a good description of either dynamic or nondynamic correlation but few methods introduce both simultaneously in a cost-efficient manner. Therefore, it is important to recognize and quantify the two correlation regimes. A further distinction between DC and NDC can be made in terms of the electron density. DC only produces small (local) changes in the electron density with respect to the HF picture, whereas NDC induces large (global) changes in the electron density. The effects of electron correlation on the electron density have been used to define a separation of the correlation energy into DC and NDC energies. 33, 32, 34 We have recently used a similar strategy to separate the second-order reduced density matrix (2-RDM) into DC and NDC parts. 35 This separation has been applied to a two-electron model, providing scalar descriptors of DC and NDC in terms of natural orbital occupancies. 35

Many quantities have been constructed to analyze and quantify electron correlation 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 35 but very few provide a local account of electron correlation. In this context, we should mention the recent paper of Grimme 44 who has put forward a local NDC measure in the context of finite-temperature density functional theory (DFT), where fractional occupancies are used. 51 In the present work we aim at introducing a local correlation function that can be split into DC and NDC parts, thus providing tools for a real-space analysis of dynamic and nondynamic correlation. Upon integration over the whole Cartesian space, these new functions yield the corresponding global values recently introduced in a previous publication. 35 Furthermore, these local functions can be decomposed into orbital contributions and they only depend on natural orbitals and their occupancies. To our knowledge, this is the first scheme that provides a local account of DC and NDC contributions. These tools can be used in the context of linear-scaling algorithms to perform a real-space analysis of electron correlation. Since they depend on natural orbitals and their occupancies, they can also serve as functional ingredients 52 in density-matrix functional theories 53, 54, 55 and fractional-occupancy DFT 56, 57, 58, 59 (or regular DFT by mapping orbital occupancies into Kohn-Sham orbital energies60). These functions can also aid in designing local mixing functions 61 and attenuating functions in the context of local hybrid functionals 62, 63, 64, 65, 66, 67, 68 and range-separated functionals. 28, 69, 70, 71

2 Theory

We have recently put forward a natural-occupancy based index to account for electron correlation, IT, which can be split into DC, ID, and NDC, IND, parts 35

[TABLE]

where $n^{\sigma}_{i}$ is the occupation of the spin-natural orbital $i$ with spin $\sigma$. Notice that $I_{\text{ND}}$ obtained from this derivation 35 is actually a measure of the deviation from idempotency. 36, 37 In Ref. 35 we suggested the decomposition of the 2-RDM into dynamic and nondynamic parts. The application of the decomposition in the 2-RDM of a simple two-electron model provided the individual orbital contributions of Eq. 1-3. The final expressions were obtained upon the assumption that, for an $N$-electron system, the total measure could be retrieved from the summation of individual orbital contributions. Therefore, the indices can be regarded as sum of orbital contributions to dynamic (Eq. 2) and nondynamic (Eq. 3) correlation obtained from simplified model of the 2-RDM. By construction, these indices can be further decomposed into orbital contributions and, by multiplying each natural-orbital contribution by its natural orbital amplitude, $\left|\phi^{\sigma}_{i}(\mathbf{r})\right|^{2}$, one can transform these global expressions into local functions that account for the pertinent real-space electron correlation effects,

[TABLE]

where $\phi^{\sigma}_{i}$ is the spin-natural orbital $i$ with spin $\sigma$. Since spin-natural orbitals are normalized to 1, integration of the local measures (Eq. 4-6) over the space coordinate yields the global values for total, dynamic and nondynamic correlation, IT, ID, and IND, respectively. The latter expressions render themselves to straightforward orbital decompositions, each orbital contribution being the natural orbital amplitude weighted by a simple expression of the corresponding natural occupancy. The simplicity of these electron correlation functions make them applicable to a wide range of electronic structure methods, provided a set of natural orbitals and occupancies are available.

3 Results

All the full-configuration interaction (FCI) calculations included in this work are performed using a modified version of the program of Knowles and Handy, 72 whereas CASSCF and UHF calculations were obtained with Gaussian 09 package. 73 The local values of $I_{\text{D}}$ and $I_{\text{ND}}$ for the isoelectronic series of He and Be are shown in 1 in the form of the radial distribution of such functions (*i.e., *Eqs. 5 and 6 integrated over the solid angle). Since electron correlation can be tuned with $Z$, both systems have been previously used to study dynamic and nondynamic correlation effects. 32, 34, 35 The He and Be series have been calculated at the FCI level using two even-temperated basis sets optimized in a previous work 35 using the technique described elsewhere. 74 The He series is driven by dynamic correlation that disminishes as $Z$ increases, whereas the pseudo-degeneracy of the $2s$ and $2p$ orbitals in Be results in important nondynamic correlation effects in the Be isoelectronic series. As the value of $Z$ increases the electron density shrinks as reflected by the displacement of the local maxima of $4\pi r^{2}I_{\text{D}}(r)$ and $4\pi r^{2}I_{\text{ND}}(r)$ to smaller $r$ for both He and Be series. On the other hand, the global values, $I_{\text{D}}$ and $I_{\text{ND}}$, shown in the legend of 1, decrease with $Z$ because electron correlation is less important for contracted densities. 35 The decomposition of $4\pi r^{2}I_{\text{D}}(r)$ and $4\pi r^{2}I_{\text{ND}}(r)$ in terms of orbital contributions is given in 2. The orbital contributions to $4\pi r^{2}I_{\text{D}}(r)$ and $4\pi r^{2}I_{\text{ND}}(r)$ for the He atom, reveal that $1s$, $2s$ and $2p$ orbitals are responsible for the single peak observed in the radial distribution function of both $I_{\text{D}}(r)$ and $I_{\text{ND}}(r)$. The Be isoelectronic series shows a broad peak at large $r$ for both $4\pi r^{2}I_{\text{D}}(r)$ and $4\pi r^{2}I_{\text{ND}}(r)$, and a narrower peak at small $r$ values. The former has contributions from the $2s$ and $2p$ orbitals, while orbitals $1s$, $2s$, $3s$, and $3p$ form the peak observed in $4\pi r^{2}I_{\text{D}}(r)$ at short $r$.

The value of $r$ at which the local DC and NDC functions peak, $I_{\text{D}}(r_{\text{max}})$ and $I_{\text{ND}}(r_{\text{max}})$, increases with $Z$ for Be series, in line with the fact that the relative $2s-2p$ orbital gap decreases with $Z$. On the other hand, in the He series $I_{\text{D}}(r_{\text{max}})$ barely changes with $Z$ and $I_{\text{ND}}(r_{\text{max}})$ actually decreases with $Z$ because NDC effects are negligible in this system.

The prototypical case of nondynamic correlation occurs in bond-breaking processes. 3 and 4 include the plots of the local functions along the potential energy curves of LiH and BH at the FCI level using aug-cc-pVDZ and cc-pVDZ basis set, respectively. The global values of $I_{\text{D}}$ and $I_{\text{ND}}$ indicate that nondynamic correlation increases and the dynamic correlation decreases as the molecule stretches. The local plots show that dynamic correlation is actually more important in regions where there is more than one localized electron, such as on Li and B atoms. As both systems dissociate, the $I_{\text{D}}$ peak on H disappears because one isolated electron cannot contribute to dynamic electron correlation. On the other hand, the peak of NDC on the position of the H atom increases with $r$, a signature of the quantum entanglement occurring between the electron in H atom and another electron of opposite spin in the Li (or B) atom. In the case of BH, the NDC function presents two additional maxima at each side of the B nucleus at $r_{\text{BH}}=4.0$, which correspond to the $2p$ orbital. These additional maxima increase upon dissociation because the $2p$ orbital that was forming the sigma bond is now singly occupied and thus NDC predominates.

D2h/D4h potential energy surface (PES) of H4 has been used to assess the accuracy of electronic structure methods to account for simultaneous DC and NDC. 75, 76, 77, 78, 79 The PES, calculated at the FCI level using aug-cc-pVDZ basis set, is described by two parameters, $R$ and $\theta$ (see 5), $\theta\approx 90^{\circ}$ and large $R$ corresponding to the structures most affected by NDC. In 5 we find the isosurface plots of $I_{\text{D}}(\mathbf{r})$ and $I_{\text{ND}}(\mathbf{r})$ for $\theta=70^{\circ}$, $80^{\circ}$, and $90^{\circ}$, keeping R fixed at 0.80$\AA$. From $\theta=70^{\circ}$ to $\theta=90^{\circ}$ there is an increase of NDC due the degeneracy of b3u and b2u orbitals, 75 as confirmed by the red isosurfaces in 5. At $\theta=70^{\circ}$, $I_{\text{D}}(\mathbf{r})$ extends along two H-H bonds while $I_{\text{ND}}(\mathbf{r})$ is almost negligible. The $I_{\text{ND}}(\mathbf{r})$ increases as we move towards the $D_{4h}$ geometry, becoming larger than $I_{\text{D}}(\mathbf{r})$ at $\theta=90^{\circ}$. The local descriptors thus provide an appropriate real-space account of DC and NDC in this system.

Finally, we have also have investigated ortho-, meta- and para-benzine singlet diradicals, which have two unpaired electrons and, therefore, are affected by nondynamic correlation. An effective nondynamic localization function should identify the regions where these unpaired electrons localize, which coincide with the regions where the local spin 80, 81 is larger. 82 Natural orbitals and occupations were calculated from a CASSCF/cc-pVTZ wavefunction with 8 active electrons in 8 orbitals including six $\pi$-orbitals and two $\sigma$- orbitals at the geometries of the given level of theory. NDC arises in these systems due to the near-degeneracy of the HOMO and LUMO $\sigma$-orbitals, which are localized at the position of the radical centers. The three isomers have an $I_{\text{ND}}(\mathbf{r})$ localized at the radical positions, showing the regions of the molecule where NDC is important (see 6). $I_{\text{D}}(\mathbf{r})$ is delocalized along the $\pi-$orbitals and only has some small contributions from the $\sigma$-orbitals for ortho- and meta-benzine.

To illustrate that the local correlation functions introduced in this work can be applied to any electronic structure method providing a set of orbitals and fractional occupations, we have computed $I_{\text{D}}(\mathbf{r})$ and $I_{\text{ND}}(\mathbf{r})$ for para-benzine at the UHF/cc-pVTZ level. Some NDC can be captured in a HF wavefunction by breaking the spin symmetry at the expense of not being an eigenfunction of $\hat{S}^{2}$. The unrestricted natural orbitals (UNO) 83, 84 are often used to generate starting orbitals for MCSCF/CASSF wavefunctions computed in regions of the potential energy surface where nondynamic electron correlation is dominant. The UNOs are obtained by diagonalization of the transformed UHF charge density matrix. 83 The UNO occupancies can be input in Eq. 3 and 6 to assess the global and local nondynamic correlation obtained from the UHF wavefunction. We can observe in 7 that $I_{\text{ND}}(\mathbf{r})$ isosurfaces are qualitatively equivalent for UNO and CASSCF(8,8) natural orbitals indicating than both methods, UHF and CASSCF, capture NDC in the region where the unpaired electrons are localized in the molecule.

In principle, the UNOs and their occupancies can also be used to compute $I_{\text{D}}(r)$ (Eq.5) and $I_{\text{D}}$ (Eq.2). The total value of $I_{\text{D}}$ for para-benzine at the UHF level is 0.60, which is quite high for the UHF wavefunction that should not contain dynamic correlation. However, it is important to note that UNOs are the natural orbitals of the spin-projected UHF wave function and, as such, they should only be used to qualitatively understand nondynamic electron correlation.83, 84 In our opinion, it is unlikely that UNOs reflect the dynamic correlation contents of the UHF.

4 Conclusions

In this paper we have introduced real-space descriptors of dynamic and nondynamic electron correlation. Theses local functions can be constructed from any electronic structure calculation provided a set of orbitals and occupation numbers are available. The functions admit orbital decomposition and, upon integration over the Cartesian space, yield the corresponding scalar descriptors recently proposed in Ref 35. The local correlation functions have been computed for a series of systems with varying electron correlation regimes, showing their ability to differentiate regions of the molecule where dynamic or nondynamic correlation are important. The DC and NDC local functions presented in this work hold the promise to aid in the development of new local correlation methods and hybrid methods.

Acknowledgements

The authors thank Prof. Ludeña and Prof. Salvador for insightful discussions. This research has been funded by Spanish MINECO/FEDER Project No. CTQ2014-52525-P and the Basque Country Consolidated Group Project No. IT588-13. Technical and human support provided by IZO-SGI, SGIker (UPV/EHU, MICINN, GV/EJ, ERDF and ESF) is gratefully acknowledged. ERC acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement (No. 660943).