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Developing orbital-free quantum crystallography: the local potentials and associated partial charge densities

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aQuantum Chemistry, D.I. Mendeleev University, Miusskaya Sq. 9, Moscow, 125047, Russian Federation, and bA.N. Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, 28 Vavilova str., Moscow, 119991, Russian Federation
*Correspondence e-mail: vtsirelson@yandex.ru

Edited by P. Macchi, Politecnico di Milano, Italy (Received 11 April 2021; accepted 30 May 2021; online 16 July 2021)

This work extends the orbital-free density functional theory to the field of quantum crystallography. The total electronic energy is decomposed into electrostatic, exchange, Weizsacker and Pauli components on the basis of physically grounded arguments. Then, the one-electron Euler equation is re-written through corresponding potentials, which have clear physical and chemical meaning. Partial electron densities related with these potentials by the Poisson equation are also defined. All these functions were analyzed from viewpoint of their physical content and limits of applicability. Then, they were expressed in terms of experimental electron density and its derivatives using the orbital-free density functional theory approximations, and applied to the study of chemical bonding in a heteromolecular crystal of ammonium hydro­oxalate oxalic acid dihydrate. It is demonstrated that this approach allows the electron density to be decomposed into physically meaningful components associated with electrostatics, exchange, and spin-independent wave properties of electrons or with their combinations in a crystal. Therefore, the bonding information about a crystal that was previously unavailable for X-ray diffraction analysis can be now obtained.

1. Introduction

The position-space distribution of the kinetic energy of electrons in the N-electron many-nuclear system is determined by the spread of the permissible values of the momentum of each electron, dictated by the laws of quantum mechanics and by volume, which confines the electron motion. This volume is defined by the nuclei configuration and by distribution of the remaining (N − 1) electrons. In addition, the momentum of an electron and the length of the wave associated with it are related in the position space (de Broglie, 1924[Broglie, L. (1924). London Edinb. Dubl. Philos. Mag. J. Sci. 47, 446-458.]). The Heisenberg (1927[Heisenberg, W. (1927). Z. Phys. 43, 172-198.]) uncertainty principle links the uncertainty in the position of an electron with the spread of allowed values of its momentum, i.e. with the inherent quantum fluctuations (Kirzhnits et al., 1975[Kirzhnits, D. A., Lozovik, Yu. E. & Shpatakovskaya, G. V. (1975). Sov. Phys. Usp. 18, 649-672.]; Ludeña, 1982[Ludeña, E. V. (1982). J. Chem. Phys. 76, 3157-3160.]; Nelson, 1985[Nelson, E. (1985). Quantum Fluctuations. Princeton University Press.]; Hamilton et al., 2007[Hamilton, I. P., Mosna, R. A. & Delle Site, L. (2007). Theor. Chem. Acc. 118, 407-415.]). As a result, standing electronic waves with allowed values of (quasi)momentum are formed. These waves build the discrete quantum energy levels that satisfy the Pauli exclusion principle and manifest themselves in the position space as electron shells corresponding to different values of the principal quantum number (Pauling, 1927[Pauling, L. (1927). Proc. R. Soc. A114, 181-211.]; Unsold, 1927[Unsold, A. (1927). Ann. Phys. 387, 355-393.]; Hellmann, 1937[Hellmann, H. (1937). Einfuhrung in die Quantenchemie. Liepzig and Vienna: Franz Deuticke.]; Weisskopf, 1975[Weisskopf, V. F. (1975). Science, 187, 605-612.]).

The spread of electron momentum values in different directions of the position space depends on the specific crystal structure, so the electron shells in multinuclear systems are nonspherical with respect to the nuclei. As a result, an inhomogeneous distribution of the electron density of a crystal, ρ(r), is formed. This is what determines a phenomenon called chemical bonding.

The question arises: how to describe the nature of the chemical bonding in all its diverse manifestations, relying mainly on the above physical-based picture of the spatial distribution of electrons and their energy characteristics (potentials), and without resorting to ill-defined orbital representations? In this paper, we develop an approach proposed by Tsirelson & Stash (2020[Tsirelson, V. & Stash, A. (2020). Acta Cryst. B76, 769-778.]), which solves this problem. The method is based on the orbital-free density functional theory (DFT) (Wesolowski & Wang, 2013[Wesolowski, T. A. & Wang, Y. A. (2013). Recent Progress in Orbital-free Density Functional Theory. World Scientific.]; Witt et al., 2018[Witt, W. C., del Rio, B. G., Dieterich, J. M. & Carter, E. A. (2018). J. Mater. Res. 33, 777-795.]; Nagy, 2018[Nagy, A. (2018). In Many-body Approaches at Different Scales, edited by G. G. N. Angilella & C. Amovilli, pp. 253-260. Springer International Publishing AG.]). We analyze the one-electron potentials and associated partial electron densities assuming that total ρ(r) is known from a precise X-ray diffraction experiment and, therefore, both the electrostatic and quantum effects are incorporated into it. The density ρ(r) is presented in the form of the space-distributed structural multipole model (Hansen & Coppens, 1978[Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921.]; Tsirelson & Ozerov, 1996[Tsirelson, V. G. & Ozerov, R. P. (1996). Electron Density and Bonding in Crystals. Bristol and Philadelphia: Institute of Physics Publishing.]): a superposition of aspherical atomic densities, each of which is expanded into a convergent series over real spherical harmonics (multipoles). The Fourier transformation of model density yields the model structure factors. The electronic populations of atomic multipoles as well as their contraction/expansion parameters are determined by the least-squares fit of these structure factors to the X-ray diffraction experimental ones.

It is important that experimental electron density, ρ(r) > 0, by derivation obeys the normalization condition ∫ρ(r)dr = N (Tsirelson & Ozerov, 1996[Tsirelson, V. G. & Ozerov, R. P. (1996). Electron Density and Bonding in Crystals. Bristol and Philadelphia: Institute of Physics Publishing.]) and, therefore, it is N-representable (Gilbert, 1975[Gilbert, T. L. (1975). Phys. Rev. B, 12, 2111-2120.]). This ρ(r) is also pure-state v-representable for actual systems of interest, i.e. it corresponds to existing external potentials (Hohenberg & Kohn, 1964[Hohenberg, P. & Kohn, W. (1964). Phys. Rev. 136, B864-B871.]; Levy, 1979[Levy, M. (1979). Proc. Natl Acad. Sci. USA, 76, 6062-6065.]; Herring, 1986[Herring, C. (1986). Phys. Rev. A, 34, 2614-2631.]; Herring & Chopra, 1988[Herring, C. & Chopra, M. (1988). Phys. Rev. A, 37, 31-42.]). Therefore, it is possible to write the Euler equation for the root square of density via physically meaningful kinetic and static electronic potentials and to express them in terms of experimental density by using various orbital-free DFT approximations (Kohanoff, 2006[Kohanoff, J. (2006). In Electronic Structure Calculations for Solids and Molecules: Theory and Computational Methods, pp. 75-120. Cambridge: Cambridge University Press.]; Tsirelson, 2007[Tsirelson, V. G. (2007). The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, edited by C. Matta and R. Boyd, pp. 259-283. Wiley-VCH, Weinheim.]; Wesolowski & Wang, 2013[Wesolowski, T. A. & Wang, Y. A. (2013). Recent Progress in Orbital-free Density Functional Theory. World Scientific.]; Karasiev et al., 2014[Karasiev, V. V., Chakraborty, D. & Trickey, S. B. (2014). In Many-Electron Approaches in Physics, Chemistry and Mathematics, edited by V. Bach & L. Delle Site, pp. 113-134. Cham, Switzerland: Springer.]). In addition, we avoid the variational solution of the Euler equation and analyze immediately the mentioned potentials. It opens an alternative pathway for the elucidation of the nature of atomic and molecular interactions in solids.

2. Methodology

The Kohn–Sham potential (Kohn & Sham, 1965[Kohn, W. & Sham, L. J. (1965). Phys. Rev. 140, A1133-A1138.]) acting on noninteracting electrons yields the ground-state density of a real system described by Kohn–Sham orbitals. Orbital-free DFT is based on the approximations to Kohn–Sham DFT equations without using the Kohn–Sham orbitals. Let us start from the Euler equation for an electron system in the stationary ground state (Gelfand & Fomin, 1963[Gelfand, I. M. & Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs, New Jersey: Prentice-Hall Inc.]; Gritsenko, van Leeuwen & Baerends, 1994[Leeuwen, R. van & Baerends, E. J. (1994). Phys. Rev. A, 49, 2421-2431.]; Wesolowski & Wang, 2013[Wesolowski, T. A. & Wang, Y. A. (2013). Recent Progress in Orbital-free Density Functional Theory. World Scientific.]):

[\mu [\rho ] = {{\delta {T}_{\rm s}[\rho ]}\over{\delta \rho }}+{v}_{\rm eff }[\rho] . \eqno(1)]

It connects the effective static one-electron potential, veff, and the functional derivative of the noninteracting kinetic energy of electrons, Ts[ρ], with respect to the electron density. A Lagrange multiplier μ follows from the condition ∫ρ(r)dr = N. It is the chemical potential μ which is commonly taken as negative of the first ionization potential (Levy et al., 1984[Levy, M., Perdew, J. P. & Sahni, V. (1984). Phys. Rev. A, 30, 2745-2748.]) and is constant for a stationary system.

The potential veff(r) is

[v_{\rm eff}({\bf r}) = -\sum _{\rm A}{ {Z_{\rm A}} \over {|{\bf r}-{\bf R}_{\rm A}|}} +{1\over{\rho ({\bf r})}}\int {{\pi ({\bf r},{\bf r}^{\prime})}\over{|{\bf r}-{\bf r}^{\prime}|}} d{\bf r}^{\prime}. \eqno(2)]

Atomic units are used throughout the article. Here, π(r,r′) is the pair electron density (McWeeny & Sutcliffe, 1969[McWeeny, R. & Sutcliffe, B. T. (1969). Methods of Molecular Quantum Mechanics. London: Academic Press.]), ZA and RA are the charge and coordinate of A nucleus, respectively. The Born–Oppenheimer approximation is assumed to be valid. Relatively small at the ground-state equilibrium, Coulomb electron correlation may be ignored for now; then the exchange-only (i.e. single-determinant) DFT version of equation (2)[link] is

[{v}_{\rm eff}({\bf r}) = -{v}_{\rm esp}({\bf r})+{v}_{x}({\bf r}). \eqno(3)]

The electron density now acts a main variable. The electrostatic potential

[{v}_{\rm esp}({\bf r}) = \sum _{\rm A}{Z}_{\rm A}/\big(|{\bf r}-{R}_{\rm A}|\big)-\int {{\rho ({{\bf r}}_{1})}\over{|{\bf r}- {{\bf r}}_{1}|}}d{\bf r}_{1} \eqno(4)]

is generated by the nuclear and electronic parts of the charge density. The electron self-interaction is canceled in equation (3)[link]. In DFT, the exchange potential is written as vx(r) = vx, hole(r) + vx, resp(r) (Gritsenko et al., 1996[Gritsenko, O., van Leeuwen, R. J. & Baerends, E. J. (1996). Int. J. Quantum Chem. 57, 17-33.]; Schipper et al., 1998[Schipper, P. R. T., Gritsenko, O. V. & Baerends, E. J. (1998). Phys. Rev. A57, 1739-1742.]). Here, vx, hole(r) < 0 is the potential resulting from the exchange-hole charge density, while vx, resp(r) > 0 describes the response of exchange screening to small density variations, when preserving the number of electrons. The last term is ∼10 times less than vx, hole in absolute value and it is not expressed via electron density (Nagy & March, 1992[Nagy, A. & March, N. H. (1992). Phys. Chem. Liq. 25, 37-42.]; Gritsenko et al., 1996[Gritsenko, O., van Leeuwen, R. J. & Baerends, E. J. (1996). Int. J. Quantum Chem. 57, 17-33.], 2016[Gritsenko, O. V., Mentel, Ł. M. & Baerends, E. J. (2016). J. Chem. Phys. 144, 204114.]; Fuentealba, 1997[Fuentealba, P. (1997). J. Phys. B At. Mol. Opt. Phys. 30, 2039-2045.]; Nagy, 2010[Nagy, A. (2010). Int. J. Quantum Chem. 110, 2117-2120.]; Kohut et al., 2016[Kohut, S. V., Polgar, A. M. & Staroverov, V. N. (2016). Phys. Chem. Chem. Phys. 18, 20938-20944.]; Giarrusso & Gori-Giorgi, 2020[Giarrusso, S. & Gori-Giorgi, P. (2020). J. Phys. Chem. A, 124, 2473-2482.]; Kreisler, 2020[Kreisler, E. (2020). Israel J. Chem. 60, 805-822.]). We also found that the local density approximation (LDA), gradient expansion approximations, generalized gradient approximation (Parr & Yang, 1989[Parr, R. G. & Yang, W. (1989). Density-Functional Theory of Atoms and Molecules. Oxford.]) and empirical approximations to vx, resp give unrealistic exchange potentials and do not provide the expected formation of electron shells. Therefore, we are forced to neglect vx, resp; it makes the potential vx(r) slightly more negative (Baerends & Gritsenko, 1997[Baerends, E. J. & Gritsenko, O. V. (1997). J. Phys. Chem. A, 101, 5383-5403.]). We computed vx(r)[= vx, hole(r)] according to Becke (1988[Becke, A. D. (1988). J. Chem. Phys. 88, 1053-1062.]).

The potential (3)[link] is simply the potential acting on an electron in a molecule (PAEM) (Yang & Davidson, 1997[Yang, Z.-Z. & Davidson, E. R. (1997). Int. J. Quantum Chem. 62, 47-53.]; Zhang et al., 2005[Zhang, M. B., Zhao, D. X. & Yang, Z. Z. (2005). J. Theor. Comput. Chem. 04, 281-288.]; Zhao & Yang, 2014a[Zhao, D. X. & Yang, Z. Z. (2014a). J. Phys. Chem. A, 118, 9045-9057.],b[Zhao, D. X. & Yang, Z. Z. (2014b). J. Comput. Chem. pp. 2014 35, 965-977.]; Bartashevich & Tsirelson, 2018[Bartashevich, E. & Tsirelson, V. (2018). J. Comput. Chem. 39, 573-580.]). It is not a functional derivative of any density functional, therefore, it is not a true effective potential (Ospadov et al., 2018[Ospadov, E., Tao, J., Staroverov, V. N. & Perdew, J. P. (2018). PNAS, 115, E11578-E11585.]). However, its component, vx, hole, shows the same electron shell-related inflection and maximum points as vx(r) in the Kohn–Sham method. Thus, replacement vx(r) by vx, hole(r) and veff(r) by vPAEM(r) preserves the properties of potentials, like the discrete electron energy levels, which we are interested in (Baerends & Gritsenko, 1997[Baerends, E. J. & Gritsenko, O. V. (1997). J. Phys. Chem. A, 101, 5383-5403.]; Martin Pendas et al., 2016[Martin Pendas, A., Francisco, E., Gallo Bueno, A., Guevara Vela, J. M. & Costales, A. (2016). In Applications of Topological Methods in Molecular Chemistry, edited by R. Chauvin, C. Lepetit., B. Silvi & E. Alikhani, ch. 6, pp. 131-150. Switzerland: Springer International Publishing.]). Also, it justifies the computation of vx, hole by using available orbital-free DFT approximations (Karasiev et al., 2014[Karasiev, V. V., Chakraborty, D. & Trickey, S. B. (2014). In Many-Electron Approaches in Physics, Chemistry and Mathematics, edited by V. Bach & L. Delle Site, pp. 113-134. Cham, Switzerland: Springer.]).

Within the presented approach, a ground-state system of interacting electrons is mapped into a system of noninteracting electrons of the same density ρ(r) and of very close topology. Potential vPAEM(r) may be seen as the negative average potential energy of any one (indistinguishable) electron belonging a given molecule and located at the point r, in the field of the remaining (N − 1) electrons and all the nuclei of a system (Sen, De Proft & Geerlings, 2002[Sen, K. D., De Proft, F. & Geerlings, P. (2002). J. Chem. Phys. 117, 4684-4693.]; Zhang et al., 2005[Zhang, M. B., Zhao, D. X. & Yang, Z. Z. (2005). J. Theor. Comput. Chem. 04, 281-288.]; Bartashevich & Tsirelson, 2018[Bartashevich, E. & Tsirelson, V. (2018). J. Comput. Chem. 39, 573-580.]; Martin Pendas et al., 2016[Martin Pendas, A., Francisco, E., Gallo Bueno, A., Guevara Vela, J. M. & Costales, A. (2016). In Applications of Topological Methods in Molecular Chemistry, edited by R. Chauvin, C. Lepetit., B. Silvi & E. Alikhani, ch. 6, pp. 131-150. Switzerland: Springer International Publishing.]). It is important to note that the minus gradient of PAEM yields the local force field, which is isomorphic to the non-gradient Ehrenfest force field (Martín Pendás & Hernández-Trujillo, 2012[Martín Pendás, A. & Hernández-Trujillo, J. (2012). J. Chem. Phys. 137, 134101.]).

Single-determinantal noninteracting kinetic energy, Ts[ρ], may be presented as a sum of the Weizsäcker (1935[Weizsäcker, C. F. (1935). Z. Phys. 96, 431-434.]) and Pauli (Herring, 1986[Herring, C. (1986). Phys. Rev. A, 34, 2614-2631.]; March, 1986[March, N. H. (1986). Phys. Lett. A, 113, 476-478.], 1987[March, N. H. (1987). J. Comput. Chem. 8, 375-379.], 2010[March, N. H. (2010). J. Mol. Struct. Theochem, 943, 77-82.]; Holas & March, 1991[Holas, A. & March, N. H. (1991). Phys. Rev. A, 44, 5521-5536.]) contributions:

[T_{\rm s}[\rho] = T_{\rm W}[\rho] + T_{\rm P}[\rho]. \eqno(5)]

The Weizsäcker kinetic energy TW[ρ] = [a \int {{{{[| {\nabla \rho (\bf r)} |}^2}} / {\rho (\bf r)}}]{\rm d}{\bf r}] = [\int T_{\rm W}({\bf r}){\rm d}r] > 0 originates from the wave-particle duality of electrons and the Heisenberg uncertainty principle (Holland, 1993[Holland, P. R. (1993). The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge: Cambridge University Press.]; Hamilton et al., 2007[Alipour, M. (2015). Chem. Phys. Lett. 635, 210-212.]). It arises from a semi-local part of the quantum electron fluctuations, is expressed via the electron density at r and its gradient, and can be viewed as kinetic energy of noninteracting `spinless' particles of density ρ(r), which are in the `bosonic' ground state, when all particles are in the same lowest energy state (Liu, 2007[Liu, S. (2007). J. Chem. Phys. 126, 244103.]). For one- or two-electron systems, coefficient a is [1\over 8] and this value is used for N electrons as well, however, a may be different for the other systems (Gál & Nagy, 2000[Gál, T. & Nagy, A. (2000). J. Mol. Struct. Theochem, 501-502, 167-171.]).

The fermionic effects are not included in TW[ρ]. Meanwhile, the presence of an electron of a given spin at r affects the motion of the rest of the same-spin electrons. The Pauli energy, TP[ρ], describes the excess in the total electronic kinetic energy over the energy of the noninteracting `spinless' particles, equation (5[link]). It follows from the antisymmetry requirement for the many-electron wavefunction (Ludeña et al., 2018[Ludeña, E. V., Arroyo, D., Salazar, E. X. & Vallejo, J. (2018). Adv. Quant. Chem. 76, 59-77.]). TP[ρ] goes to zero in the regions where a single orbital populated by 1 or 2 electrons dominates, as it can be seen close to the hydrogen atom positions (see below).

Taking the functional derivative of Ts[ρ] and assuming proper homogeneity of the functionals in density scaling, we transform equation (5)[link] to

[{{\delta {T}_{\rm s}}\over{\delta \rho }} = {{\delta {T}_{\rm W}}\over{\delta \rho }}+ {{\delta {T}_{\rm P}}\over{\delta \rho }} = {{v}_{\rm W}({\bf r})+v}_{\rm P}({\bf r}) = {v}_{\rm kin}({\bf r}) \eqno(6)]

[see Liu & Ayers (2004[Liu, S. & Ayers, P. A. (2004). Phys. Rev. A, 70, 022501.]) for discussion on the existence and uniqueness of [\delta T_{\rm s} /\delta \rho ]]. vkin(r) is called the kinetic potential (King & Handy, 2000[King, R. A. & Handy, N. C. (2000). Phys. Chem. Chem. Phys. 2, 5049-5056.]). The sign-alternating Weizsäcker potential (Herring, 1986[Herring, C. (1986). Phys. Rev. A, 34, 2614-2631.]; Delle Site, 2002[Delle Site, L. (2002). Europhys. Lett. 57, 20-24.]) is

[v_{\rm W}({\bf r}) = {1 \over 8}{{{{| {\nabla \rho ({\bf r})} |}^2}} \over {{\rho ^2}({\bf r})}} - {1 \over 4}{{{\nabla ^2}\rho ({\bf r})} \over {\rho ({\bf r})}}. \eqno(7)]

Being related to the wave nature of electrons, this spin-independent potential reflects the electron-shell structure of any the many-electron system (Tsirelson et al., 2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]) and asymptotically goes to chemical potential μ at infinity. However, it may diverge asymptotically on the nodal plane of the highest-occupied orbital, if this plane exists (Gori-Giorgi & Baerends, 2018[Gori-Giorgi, P. & Baerends, E. J. (2018). Eur. Phys. J. B, 91, 160.]). This fact should be taken into account when the superposition of bound atoms, i.e. a multipole model, is used to describe the electron density.

Unfortunately, a reliable local-density approximation for potential vP(r) is unknown (Levy & Görling, 1994[Levy, M. & Görling, A. (1994). Philos. Mag. B, 69, 763-769.]; Finzel, 2016[Finzel, K. (2016). Int. J. Quantum Chem. 116, 1261-1266.]; Nagy, 2018[Nagy, A. (2018). In Many-body Approaches at Different Scales, edited by G. G. N. Angilella & C. Amovilli, pp. 253-260. Springer International Publishing AG.]; Giarrusso & Gori-Giorgi, 2020[Giarrusso, S. & Gori-Giorgi, P. (2020). J. Phys. Chem. A, 124, 2473-2482.]). Moreover, the gradient expansion of TP violates the exact condition vP(r) ≥ 0 (Levy et al., 1984[Levy, M., Perdew, J. P. & Sahni, V. (1984). Phys. Rev. A, 30, 2745-2748.]). Therefore, the expression

[{v}_{\rm P}({\bf r}) = - {v}_{\rm PAEM }({\bf r})-{v}_{\rm W}({\bf r})+\mu \eqno(8)]

is used in actual calculations (Tsirelson et al., 2013[Tsirelson, V. G., Stash, A. I., Karasiev, V. V. & Liu, S. (2013). Comput. Theor. Chem. 1006, 92-99.]; Astakhov et al., 2016[Astakhov, A. A., Stash, A. I. & Tsirelson, V. G. (2016). Int. J. Quantum Chem. 116, 237-246.]; Gritsenko et al., 2016[Gritsenko, O. V., Mentel, Ł. M. & Baerends, E. J. (2016). J. Chem. Phys. 144, 204114.]; Śmiga et al., 2020[Śmiga, S., Siecińska, S. & Fabiano, E. (2020). Phys. Rev. B, 101, 165144.]).

Bringing together all the local potentials, we arrive at the Euler equation in the form

[\mu = {v}_{\rm kin}({\bf r})+{{v}_{\rm PAEM }({\bf r})}. \eqno(9)]

Here, a static potential vPAEM(r) = −vesp(r) + vx(r) includes the components that depend on the equilibrium nuclear configuration and on corresponding electron density. Kinetic potential vkin(r) = vP(r) + vW(r) contains the terms related with electron motion.

Equation (9)[link] is central in our consideration. First, it allows the presentation of the kinetic and static potentials as a sum of various local potentials reflecting definite physical effects. Second, the complete bonding picture is now contained in a small set of one-particle potentials; therefore, they provide the compact basis for the position-space chemical bonding analysis in terms of potential barriers, steps and wells (Gritsenko et al., 1994[Gritsenko, O., van Leeuwen, R. & Baerends, E. J. (1994). J. Chem. Phys. 101, 8955-8963.]). Third, equation (9)[link] shows that for the stationary ground state, the one-electron kinetic potential vkin is strictly linked with PAEM: it is equal to −vPAEM + μ (Liu & Ayers, 2004[Liu, S. & Ayers, P. A. (2004). Phys. Rev. A, 70, 022501.]; Tsirelson & Stash, 2020[Tsirelson, V. & Stash, A. (2020). Acta Cryst. B76, 769-778.]; Finzel & Bultinck, 2020[Finzel, K. & Bultinck, P. (2020). Acta Phys. Chim. Sin. 34, 650-655.]; Finzel & Kohout, 2018[Finzel, K. & Kohout, M. (2018). Theor. Chem. Acc. 137, 182.]; Finzel, 2018[Finzel, K. (2018). Comput. Theor. Chem. 1144, 50-55.], 2019[Finzel, K. (2019). J. Chem. Phys. 151, 024109.]; 2020[Finzel, K. (2020). Molecules, 25, 1771.]).

We cannot define the constant μ for an actual crystal within the orbital-free scheme. Therefore, we suggested for this purpose the empirical formula [\mu = \sum _{\rm k}{\mu }_{\rm k }N_{\rm k} /\sum _{\rm k}N_{\rm k}] (k numerates the atoms in a unit cell; Nk is the number of electrons in atom k), which underestimates the chemical potential by ∼10%. Such uncertainty is within the error in μ values computed by different methods (Gritsenko et al., 2016[Gritsenko, O. V., Mentel, Ł. M. & Baerends, E. J. (2016). J. Chem. Phys. 144, 204114.]).

Combining the orbital-free treatment of one-electron potentials with QTAIM (Bader, 1990[Bader, R. F. W. (1990). Atoms in Molecules: A Quantum Theory. New York: Oxford University Press.]), we recently presented the topological analysis of the kinetic and PAEM potential and related force fields in a crystal (Tsirelson & Stash, 2020[Tsirelson, V. & Stash, A. (2020). Acta Cryst. B76, 769-778.]). Now we are in position to make one more step and to elaborate the concept of the partial charge densities, which removes the problem of the chemical potential choice. It is known that exchange electron density and corresponding potential may be connected via the Poisson equation ∇2vx(r) = −4πqx(r) (Harbola & Sahni, 1989[Harbola, M. K. & Sahni, V. (1989). Phys. Rev. Lett. 62, 489-492.]; Liu et al., 1999[Liu, S., Ayers, P. W. & Parr, R. G. (1999). J. Chem. Phys. 117, 6197-6203.]; Goerling, 1999[Goerling, A. (1999). Phys. Rev. Lett. 83, 5459.]; March, 2002[March, N. H. (2002). Phys. Rev. A, 65, 034501.]; Sen et al., 2002[Sen, K. D., De Proft, F. & Geerlings, P. (2002). J. Chem. Phys. 117, 4684-4693.]; Tsirelson et al., 2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]). We will extend this idea and apply the Laplace operator to the left and right parts of equation (9)[link]:

[{\nabla }^{2}{v}_{\rm kin}({\bf r}) = -{\nabla }^{2}{{v}_{\rm PAEM }({\bf r})}. \eqno(10)]

The Poisson equations for these potentials and associated densities are

[\eqalign{{\nabla }^{2}{v}_{\rm PAEM }({\bf r}) &= -4\pi {q}_{\rm PAEM}({\bf r}), \cr {\nabla ^2}{v_{\rm kin}}({\bf r}) &= - 4\pi {q_{\rm kin}}({\bf r}).} \eqno(11)]

From equations (11)[link], there follows a link between two charge densities: qkin(r) = −qPAEM(r). Further, in the spirit of the works by Tsirelson et al. (2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]) and Mi et al. (2018[Mi, W., Genova, A. & Pavanello, M. (2018). J. Chem. Phys. 148, 184107.]), we decompose functions qPAEM(r) and qkin(r) into the partial charge densities qi(r), corresponding to potentials vi(r): ∇2vi(r) = −4πqi(r). Index i = esp, x, P, W stands here for electrostatic, exchange, Pauli and Weizsäcker components, respectively. As a result, we have

[\eqalign{&{\nabla }^{2}{v}_{\rm PAEM}({\bf r}) = -4\pi {q}_{\rm PAEM}({\bf r})\, \rightarrow \cr &{{\nabla }^{2}\{-v}_{\rm esp}({\bf r})+{v}_{\rm x}({\bf r})\} = -4\pi {\{-q}_{\rm esp}({\bf r})+{q}_{\rm x}({\bf r})\},} \eqno(12)]

[\eqalign{&{\nabla }^{2}{v}_{\rm kin}({\bf r}) = -4\pi {q}_{\rm kin}({\bf r}) \, \rightarrow \cr &{{\nabla }^{2}\{v}_{\rm W}({\bf r})+{v}_{\rm P}({\bf r})\} = -4\pi {\{q}_{\rm W}({\bf r})+{q}_{\rm P}({\bf r})\}.} \eqno(13)]

The constant μ is no longer present in this description.

Ultimately, each partial charge density, qi(r), presents the contribution accounting for proper physical effect. It allows the potentials to be analysed by considering the extremes and inflection points of qi(r). It leads to the decomposition of the total electron density similar to the proposed decomposition of the total energy (Liu, 2007[Liu, S. (2007). J. Chem. Phys. 126, 244103.]). The partial charge densities are associated with electrostatics, exchange, and spin-independent wave properties of electrons or with their combinations. Each partial density is computed from appropriate potentials which, in turn, are expressed via experimental electron density. Therefore, the solution of the Euler equation becomes now unnecessary.

In this work, we applied the approach outlined above to consider chemical bonding in a heteromolecular crystal of ammonium hydro­oxalate oxalic acid dihydrate (I), NH4+·C2HO4·C2H2O4·2H2O (see Fig. 1[link]) in which chemical bonds of different types coexist. We used the results of our high-resolution synchrotron-radiation diffraction study at 15 K (Stash et al., 2013[Stash, A. I., Chen, Y. S., Kovalchukova, O. V. & Tsirelson, V. G. (2013). Russ. Chem. Bull. 62, 1752-1763.]). The electrostatic potential is computed from experimental electron density according equation (4)[link] and potential vx(r) is taken from Becke (1988[Becke, A. D. (1988). J. Chem. Phys. 88, 1053-1062.]) approximation. Potentials vP and vW are calculated via equations (7)[link] and (8)[link]. Corresponding partial charge densities were calculated numerically. The software WinXPRO (version 4.3.40; Stash & Tsirelson, 2014[Stash, A. I. & Tsirelson, V. G. (2014). J. Appl. Cryst. 47, 2086-2089.], 2020[Stash, A. & Tsirelson, V. (2020). WinXPRO, 3DPlot and TrajPlot: Computer Software for Orbital-free Quantum Crystallography. Quantum Crystallography Online Meeting, 27-29 August 2020. P.34. CentraleSupelec, France.]) was used.

[Figure 1]
Figure 1
Ammonium hydro­oxalate oxalic acid dihydrate (I): the atomic structure and fragment (outlined with a rectangle) containing the bonds which are analyzed in this work.

3. Results and discussion

We aim to reveal how the chemical bonding features may be described in terms of potentials and its associated partial densities. We begin our consideration from the C1—C2 single bond (1.550 Å) (Fig. 2[link]) and the C1=O2 double bond (1.222 Å) (Figs. 2[link] and 3[link]) of compound (I).

[Figure 2]
Figure 2
Potentials (a) and partial charge densities (b) along the C1—C2 bond in (I). Here and hereinafter atomic units (a.u.) are used.
[Figure 3]
Figure 3
Potentials (a) and partial charge densities (b) along the C1=O2 bond in (I).

First of all, we stress that neither vkin(r) nor vPAEM(r) potentials show distinguishable features of the electronic shells. It makes the potential decomposition, discussed in this paper, especially desirable in the bonding analysis. The classic component of vPAEM(r), the electrostatic potential, is linked to the total charge density [\textstyle\sum _{\rm A} Z_{\rm A}\delta ({\bf r} - R_{\rm A}) - \rho ({\bf r})], which includes both nuclear and electron contributions (Sen & Politzer, 1989[Sen, K. D. & Politzer, P. (1989). J. Chem. Phys. 90, 4370-4372.]; Leboeuf et al., 1999[Leboeuf, M., Köster, A. M., Jug, K. & Salahub, D. R. (1999). J. Chem. Phys. 111, 4893-4905.]; Roy et al., 2009[Roy, D. K., Balanarayan, P. & Gadre, S. R. (2009). J. Chem. Sci. 121, 815-821.]; Tsirelson et al., 2001[Tsirelson, V., Avilov, A., Lepeshov, G., Kulygin, A., Stahn, J., Pietsch, U. & Spence, J. C. H. (2001). J. Phys. Chem. B, 105, 5068-5074.]; Politzer & Murray, 2002[Politzer, P. & Murray, J. S. (2002). Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chim Acta), 108, 134-142.]). Electrons are bound by the negative of vesp(r): function −vesp(r) grows from the nuclear positions to flat maximum at the middle of C—C bonds and does not reveal the electronic shells (Fig. 2[link]a). Note that potentials vPAEM(r) and −vesp(r) closely resemble each other.

The quantum component of vPAEM(r), the exchange potential vx(r) < 0, approximated according to Becke (1988[Becke, A. D. (1988). J. Chem. Phys. 88, 1053-1062.]), exhibits deep negative wells at the nuclei of C atoms (Fig. 2[link]a). Also, the local negative maxima between the K and L shells and a bond midpoint plateau between atoms are observed in agreement with results of the others (Baerends & Gritsenko, 1997[Baerends, E. J. & Gritsenko, O. V. (1997). J. Phys. Chem. A, 101, 5383-5403.]; Bartashevich & Tsirelson, 2018[Bartashevich, E. & Tsirelson, V. (2018). J. Comput. Chem. 39, 573-580.]; Levina et al., 2021[Levina, E. O., Khrenova, M. G. & Tsirelson, V. G. (2021). J. Comput. Chem. 42, 870-882.]; Shteingolts et al., 2021[Shteingolts, S. A., Stash, A. I., Tsirelson, V. G. & Fayzullin, R. R. (2021). Chem. A Eur. J, 27, 7789-7809.]). These local maxima become less evident when LDA is used for vx(r). Note that the mentioned plateau, the location of σ-bond electrons, coincides with the inner part of the exchange hole which is located between bound C atoms.

Partial density qx(r) makes the effect of electronic exchange more visible. The C—C bond midpoint plateau in vx(r) (Fig. 2[link]b) corresponds to the locally positive density qx(r). This function changes its sign between K and L electronic shells marking the barriers of the exchange potential, which prevent the electron concentration in these regions. vx(r) is strongly negative in the vicinity of the nuclei in agreement with theory (Liu et al., 1999[Liu, S., Ayers, P. W. & Parr, R. G. (1999). J. Chem. Phys. 117, 6197-6203.]; Goerling, 1999[Goerling, A. (1999). Phys. Rev. Lett. 83, 5459.]; Ayers & Levy, 2001[Ayers, P. & Levy, M. (2001). J. Chem. Phys. 115, 4438-4443.]), which claims that density qx(r) must obey the exact sum rule ∫qx(r)dr = −1 (the integral is taken over the entire system or over the zero-flux atomic basins).

The kinetic potential vkin(r) is a significant component of the Euler equation (9)[link] (Jacob & Neugebauer, 2014[Jacob, C. R. & Neugebauer, J. (2014). WIREs Comput. Mol. Sci. 4, 325-362.]; Astakhov et al., 2016[Astakhov, A. A., Stash, A. I. & Tsirelson, V. G. (2016). Int. J. Quantum Chem. 116, 237-246.]; Mi et al., 2018[Mi, W., Genova, A. & Pavanello, M. (2018). J. Chem. Phys. 148, 184107.]; Tsirelson & Stash, 2020[Tsirelson, V. & Stash, A. (2020). Acta Cryst. B76, 769-778.]). Nevertheless, its chemical interpretation is little studied. We have already indicated that Weizsäcker kinetic density, tW(r) = [{{1}\over{8}} [{{{|\nabla \rho ({\bf r})|}^{2}}/{\rho ({\bf r})}}]], reflects the wave-particle spin-independent behavior of electrons in a system as well as a manifestation of the uncertainty principle. Density tW(r) is proportional to the position-space Fisher information density i(r) = [{ {|\nabla \rho ({\bf r}) |}^{2}} / {\rho ({\bf r})}], which measures spatial localization of electrons (Hamilton et al., 2007[Hamilton, I. P., Mosna, R. A. & Delle Site, L. (2007). Theor. Chem. Acc. 118, 407-415.]; Tsirelson et al., 2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]; Dehesa et al., 2011[Dehesa, J. S., Esquivel, R. O., Plastino, A. R. & Sanchez-Moreno, P. (2011). J. Russ. Laser Res. 32, 403-411.]; Alipour & Mohajeri, 2012[Alipour, M. & Mohajeri, A. (2012). Chem. Phys. 392, 105-106.]; Alipour, 2015[Alipour, M. (2015). Chem. Phys. Lett. 635, 210-212.]). The latter, in turn, is proportional to the nonclassical variance of the conjugate electron momentum and, therefore, gives the allowed range of quantum fluctuations of the electron momentum in the coordinate representation (Hall, 2000[Hall, M. J. W. (2000). Phys. Rev. A, 62, 012107.]). The local values of the kinetic energy of electrons arise from the square of the local fluctuation of the momentum operator; they are described by the distribution of allowed electronic wavenumbers, [ - {1 \over 2}[{{| {\nabla \rho ({\bf r})} |} / {\rho ({\bf r})}}]] (Kohout et al., 1991[Kohout, M., Savin, A. & Preuss, H. (1991). J. Chem. Phys. 95, 1928-1942.]; Nagy & March, 1997[Nagy, A. & March, N. H. (1997). Mol. Phys. 90, 271-276.]; Bohórquez & Boyd, 2008[Bohórquez, H. J. & Boyd, R. (2008). J. Chem. Phys. 129, 024110.]; Nagy et al., 2013[Nagy, A., Romera, E. & Liu, S. B. (2013). Phys. Lett. A, 377, 286-290.]). Interference of electronic waves forms the electronic shells. In free atoms, they manifest themselves as the spherical spatial maxima of the electron momentum density in r representation. The shell boundaries coincide with the inflection points located after each local maximum.

The electrons, bound by Weizsäcker potential vW(r), are confined within some space volume (Hellmann, 1937[Hellmann, H. (1937). Einfuhrung in die Quantenchemie. Liepzig and Vienna: Franz Deuticke.]; Ospadov et al., 2018[Ospadov, E., Tao, J., Staroverov, V. N. & Perdew, J. P. (2018). PNAS, 115, E11578-E11585.]). In molecules and solids, its topology depends of the specific atomic surrounding. For example, potential vW(r) variates its sign along the C—C bond in (I) (Fig. 2[link]a). In the regions of positive vW(r), an electron motion is classically allowed from the energy viewpoint; corresponding vW(r) maxima are observed around the midpoint of the C—C covalent bond and in atomic cores. The K and L shells are separated by the negative vW(r) minima (wells), where electrons exhibit quantum behavior (Hunter, 1975[Hunter, G. (1975). Int. J. Quantum Chem. 9, 237-242.], 1986[Hunter, G. (1986). Int. J. Quantum Chem. 29, 197-204.]; Kohout, 2001[Kohout, M. (2001). Int. J. Quantum Chem. 83, 324-331.]).

On the other hand, the first term in equation (7)[link] is always positive, whereas the second term depends on sign of ∇2ρ(r). Therefore, the positive regions of vW(r) correspond to regions of negative Laplacian of the electron density, an empirical indicator of electron concentration in Bader's (1990[Bader, R. F. W. (1990). Atoms in Molecules: A Quantum Theory. New York: Oxford University Press.]) QTAIMC theory. The negative values of vW(r) fall in the regions of electron density depletion. However, Sagar et al. (1988[Sagar, R. P., Ku, A. C. T. &, Jr, V. H. (1988). Can. J. Chem. 66, 1005-1012.]) indicated that regions of ∇2ρ(r) < 0 are places of charge concentration, rather than regions of space which are energetically allowed for electrons from the classical mechanics viewpoint. Therefore, the electron-density and energy-density approaches may give different information.

The partial charge density qW(r) may be both positive and negative because of the following property: [{\sum }_{{\Omega }_{i}}{\int }_{{\Omega }_{i}}^{ }{q}_{\rm W}({\bf r})d{\bf r} = 0] (Tsirelson et al., 2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]; Liu et al., 2018[Liu, S., Liu, L., Yu, D., Rong, C. & Lu, T. (2018). Phys. Chem. Chem. Phys. 20, 1408-1420.]). That means the density qW(r) has the nodes in position space, where its contribution to ρ(r) is zero. The specific qW(r) behavior depends on the compound under study. At the middle of the C—C distance in (I), qW(r) shows a small local negative area. Also, regions with qW(r) < 0 are placed in the valence L shells and atomic K cores. These regions are the ones of total electron density decrease due to the wave-particle property of electrons and due to space restrictions imposed by the uncertainty principle.

Pauli potential, vP(r), which accounts for manifestation of Fermi–Dirac statistics, removes electrons away from regions where other electrons of the same spin are present. Also, it implicitly contains the contribution from variations of the response density (Levy et al., 1984[Levy, M., Perdew, J. P. & Sahni, V. (1984). Phys. Rev. A, 30, 2745-2748.]; Buijse et al., 1989[Buijse, M. A., Baerends, E. J. & Snijders, J. G. (1989). Phys. Rev. A, 40, 4190-4202.]; Gritsenko et al., 1994[Gritsenko, O., van Leeuwen, R. & Baerends, E. J. (1994). J. Chem. Phys. 101, 8955-8963.]). vP(r) attains a finite value at the nuclear positions and decays in the (multipole) pseudoatoms to zero with r → ∞ (Levy & Ou-Yang, 1988[Levy, M. & Ou-Yang, H. (1988). Phys. Rev. A, 38, 625-629.]). Along the C—C bond in (I), Pauli potential flatly contributes to kinetic energy density in mid-bond region and shows even more positive stepwise behavior between the electron shells as predicted by Bartolotti & Acharya (1982[Bartolotti, L. J. & Acharya, P. K. (1982). J. Chem. Phys. 77, 4576-4585.]). In addition, density qP(r) exhibits a positive spherical barrier around each nucleus with a radius of ∼ 0.24 Å. These barriers clearly denote the boundaries between K and L electronic shells of bound carbon atoms, where electron density is depleted. Along the C—C bond line, Pauli potential is flat and small; it promotes here the electron density accumulation.

The associated Pauli charge density, qP(r) (Fig. 2[link]b), also has position-space nodes and clearly shows the electron shell structure and the other features of the C—C bond. It exhibits positive values at the nuclear positions, and alternative areas along the C—C covalent bond. Note that positive intershell peaks of qP(r) coincide with maxima in vP.

Inspection of the potential and associated density profiles along the C1=O2 double bond in compound (I) (Fig. 3[link]a), shows that the general pattern in the vicinity of the C atom is not visibly changed as compared with the C—C bond (Fig. 2)[link]. However, the picture differs drastically closer to the O atom, which more strongly attract electrons. Here, the Pauli potential exhibits a higher barrier located at 0.33 Å from the O atom position. It shows that the intershell kinetic repulsion region became even less attainable here for the same-spin electrons. It is interesting that the electrostatic potential exhibits only small asymmetry along the C—O bond. Therefore, electrostatics is not a dominating factor governing the electron density shift to O; it rather results from the joint actions of other factors. Thus, the potentials are able to distinguish between polar and nonpolar chemical bonds and may serve as a reliable indicator of the bonding nature of a system.

The partial density profiles (Fig. 3[link]b) reveal that contributions from functions qP(r) and qW(r) almost compensate each other along the C—O line; however, the former is more noticeable near the O nuclei, and the second prevails between the core and valence electron shells of this atom.

Here we need to discuss the following circumstance. The components of electronic energy at r are linked in orbital-free DFT to associated potentials by the expression, i = esp, x, P, W, provided all proper scaling conditions are satisfied (King & Handy, 2000[King, R. A. & Handy, N. C. (2000). Phys. Chem. Chem. Phys. 2, 5049-5056.]; Trickey et al., 2009[Trickey, S. B., Karasiev, V. V. & Jones, R. S. (2009). Int. J. Quantum Chem. 109, 2943-2952.]; Borgoo & Tozer, 2013[Borgoo, A. & Tozer, D. J. (2013). J. Chem. Theory Comput. 9, 2250-2255.]). Therefore, the negative potentials −vesp(r) and vx(r) should lead to attractive (stabilizing) local potential energy and to contribute to the formation of a bound ground state. The more negative the potential vx(r) is along the bond, the more easily electrons are shared between bound atoms and, therefore, the more significant is the covalence contribution to the bond between these atoms. Correspondingly, the positive Pauli potential results in positive (repulsive, destabilizing) local contributions to the energy density in agreement with exact consideration (Levy et al., 1984[Levy, M., Perdew, J. P. & Sahni, V. (1984). Phys. Rev. A, 30, 2745-2748.]).

As opposed to the above, the kinetic Weizsäcker potential takes both positive and negative values; this leads to local violation of the exact condition: tW(r) = [{{{|\nabla \rho ({\bf r})|}^{2}}/[{8\rho ({\bf r})}}]] > 0 (Gál & Nagy, 2000[Gál, T. & Nagy, A. (2000). J. Mol. Struct. Theochem, 501-502, 167-171.]; Ayers et al., 2002[Ayers, P., Parr, R. G. & Nagy, A. (2002). Int. J. Quantum Chem. 90, 309-326.]; Trickey et al., 2009[Trickey, S. B., Karasiev, V. V. & Jones, R. S. (2009). Int. J. Quantum Chem. 109, 2943-2952.]). According to Hunter (1975[Hunter, G. (1975). Int. J. Quantum Chem. 9, 237-242.]), the negative vW(r) minima should indicate the regions where electrons exhibit quantum behavior and local electron energy should take here the negative values, violating the positivity condition. The use of a positively defined electron kinetic energy density is justified by the fact that tkin(r) = tW(r) + TP(r) is consistent with a set of positivity properties for a reasonable decomposition of the Tkin[ρ] functional. At the same time, we know that kinetic energy density tW(r) is nonunique: it is described by a function defined up to the Laplacian of the electron density multiplied by an arbitrary coefficient (Ayers et al., 2002[Ayers, P., Parr, R. G. & Nagy, A. (2002). Int. J. Quantum Chem. 90, 309-326.]; García-Aldea & Alvarellos, 2007[García-Aldea, D. & Alvarellos, J. E. (2007). J. Chem. Phys. 127, 144109.]; Anderson et al., 2010[Anderson, J. S. M., Ayers, P. W. & Rodriguez Hernandez, J. I. (2010). J. Phys. Chem. 114, 8884-8895.]). The Laplacian integrates to zero and the integral value of tW[ρ] stays strictly positive. Thus, the physical meaning of the integrand cannot be always understood consistently. This fact, plus the arbitrary choice of coefficient 1/8 (see above), leads to the conclusion that the interpretation of potential vW(r) must be made with care. Nevertheless, the property: [{\sum }_{{\Omega }_{i}}{\int }_{{\Omega }_{i}}^{ }{q}_{\rm W}({\bf r}){\rm d}{\bf r}] = 0 (Tsirelson et al., 2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]) makes the Weizsäcker charge density a useful tool for extraction of relevant quantum information from the electron density.

Now let us analyze the hydrogen bonds of different strength: O3⋯H5 (1.428 Å) and O4⋯H92 (1.739 Å). Corresponding potentials and partial densities are shown in Figs. 4[link] and 5[link], respectively. In the short (strong) hydrogen-bond O3⋯H5, these functions in the vicinity of the O atom behave just like those in the C—O bond. However, a very different picture is observed closer to the H-atom position. The Pauli potential and associated density exhibit here the barrier sited at ∼0.85 Å from the O- atom position; however, both functions lose their meaning when they approach the H atom, because of the 1s orbital of H atom, dominating in this region, is populated with ∼1 electron (see explanation above). The exchange potential, vx(r), as well as vPAEM(r) do not reveal here any specific features. At the same time, qPAEM(r) yields a positive contribution to the total density along the O3⋯H5 line, which is gradually passed to the negative local contribution closer to the H atom.

[Figure 4]
Figure 4
Potentials (a) and partial charge densities (b) along the O3⋯H5 bond in (I).
[Figure 5]
Figure 5
Potentials (a) and partial charge densities (b) along the O4⋯H92 bond in (I).

The Weizsäcker potential and partial charge density have non-common profiles along the O3⋯H5 bond as compared with covalent bonds considered above. Especially, we mention the negative minimum in both functions at a distance of 0.85 Å from the O-atom position (compare with Pauli potential in this area).

The middle-strength hydrogen bond O4⋯H92 shows, generally speaking, a very similar picture. However, the Becke (1988[Becke, A. D. (1988). J. Chem. Phys. 88, 1053-1062.]) exchange potential leads here to non-smooth function qx(r) showing a singularity at a distance of 0.58 Å from the H-atom position. This artifact destroys all the other functions [see equation (8[link])] except the Weizsäcker ones. After some work, we established that this unwelcome feature results from the wrong asymptotic limit of the Becke's vx(r) for multipole-model H pseudoatoms with r → ∞ (Filippi et al., 1996[Filippi, C., Gonze, X. & Umrigar, C. (1996). In Recent Developments and Applications of Modern Density Functional Theory, edited by J. Seminario, pp. 295-326. Elsevier Science.]; Gritsenko et al., 2016[Gritsenko, O. V., Mentel, Ł. M. & Baerends, E. J. (2016). J. Chem. Phys. 144, 204114.]). The O and H multipole pseudoatomic contributions only weakly overlap at the middle of the O4⋯H92 bond; this is common for any weak noncovalent bonds or π-stacking contacts. As a result, the mentioned distortion of the potential vx(r) manifests itself in such bonds. At the same time, this artifact remains hidden in the short hydrogen bonds of (I) due to the dominating contribution from the O atom. Also, it does not appear in the vx-dependent functions for covalent bonds C1—C2, C1=O2 and O1—H1 (1.055 Å) due to the same reason. We found that a similar picture takes place for the van Leuveen & Baerends (1994[Leeuwen, R. van & Baerends, E. J. (1994). Phys. Rev. A, 49, 2421-2431.]) gradient-corrected exchange potential, while vx(r) in the LDA approximation is a suitable smooth function – see supporting information.

As far as the Weizsäcker terms are concerned, potential vW(r) when approaching the H atom shows unphysical inflection points that are immediately manifested themselves in the partial density qW(r). Thus, the H atom involved in the weak hydrogen bond is badly described within the scheme under consideration. Remembering the direct link of this potential to the kinetic density tW(r) and ambiguity in the choice of the coefficient 1/8 in expression for vW(r), we arrive at the conclusion that the interpretation of this potential for non-covalent interactions is not straightforward.

It is interesting to consider another version of the potential decomposition. According to Liu (2007[Liu, S. (2007). J. Chem. Phys. 126, 244103.]) and Tsirelson et al. (2010[Tsirelson, V. G., Stash, A. I. & Liu, S. (2010). J. Chem. Phys. 133, 114110.]), we rearrange the terms in equation (9)[link] in such way as to distinguish between the electrostatic, quantum bosonic and quantum fermionic contributions reflecting different physical effects:

[\eqalign{\mu = &-v_{\rm esp}({\bf r})+v_{\rm W}({\bf r})+v_{\rm P}({\bf r})+v_{\rm x}({\bf r})\cr =&-v_{\rm esp}({\bf r})+v_{\rm q\hbox{-}b}({\bf r})+v_{\rm q\hbox{-}f}({\bf r})} \eqno(14)]

The attractive electrostatic potential −vesp(r) is defined by equation (4)[link]. The quantum bosonic potential is exactly the Weizsäcker potential, equation (7)[link], vq-b(r) = vW(r), reflecting the quantum fluctuations of the electron momentum. The quantum fermionic potential vq-f(r) = vP(r) + vx(r) includes both static and kinetic restrictions imposed by the Fermi–Dirac statistics. This potential provides the repulsive [vP(r) dominates] or attractive [vx(r) dominates] local fermionic contributions to the electronic energy. In other words, the sign of vq-f(r) indicates which effect – the spatial static electronic exchange or kinetic Pauli factor – dominates in a given region of space. Similar decomposition is also applied to corresponding charge densities and it might be useful to link the density features to reasons of a different nature.

The 2D maps of the quantum fermionic and bosonic potentials and associated charge densities in considered fragment of (I) are given in Fig. 5[link]. All these functions reveal the electronic shells and yield a more comprehensive bonding picture than the 1D profiles. For example, they show the location and intensity of not only bonding electron pairs, but also the lone pairs. The bosonic potential in Fig. 6[link](a) shows that interference of electronic waves leads to local positive areas of potential on all the covalent bonds and to the negative ones on the hydrogen bonds of different strength. We will not discuss here the hydrogen bonds for reasons indicated above, however, we mention that the longer the hydrogen bond is, the more negative vW(r) is. Lone electronic pairs manifest themselves as heavy positive peaks of vW(r) behind the oxygen atoms. It is interesting to note the negative regions of vW(r), which appear between all atoms which are not directly bonded.

[Figure 6]
Figure 6
2D maps of the bosonic and fermionic potentials and associated charge densities in the fragment of (I): (a) vW(r), (b) qW(r), (c) vq-f(r) and (d) qq-f(r). The atomic units are used.

The associated partial bosonic density, qW(r), (Fig. 6[link]b) refines this picture. It reveals that all regions with vW(r) > 0 yield the negative contributions to the total electron density. An even more significant negative contribution comes from the regions between atoms which are not directly bonded. At the same time, the lone electron pairs behind the oxygen atoms are not seen on the qW(r) map.

The quantum fermionic potential and charge density are given in Figs. 6[link](c) and 6[link](d). We see (Fig. 6[link]d) that Pauli potential dominates on all covalent bonds, except for the C—C bond where the exchange potential is higher. The latter also dominates in the areas of location of the lone electron pairs, where the deep negative minima of vq-f(r) are observed. Partial density qq-f(r) exhibits low positive barriers on the covalent bonds and large positive barriers between atoms, which are not directly bonded. Unfortunately, it does not show the lone electron pairs of the oxygen atoms as well.

The peaks assigned to the lone electron pairs are located on vW(r) map closer to O atoms than those on the vq-f(r) map, demonstrating the fermionic effect. Their position-space configuration allows us to conclusively say that O2 and O4 atoms are in the sp2 valence state, while O1 and O3 are in the sp3 valence state.

4. Summary

After many years of intensive research, the orbital-free DFT now finds an application in various scientific fields (Wesolowski & Wang, 2013[Wesolowski, T. A. & Wang, Y. A. (2013). Recent Progress in Orbital-free Density Functional Theory. World Scientific.]; Mi et al., 2018[Mi, W., Genova, A. & Pavanello, M. (2018). J. Chem. Phys. 148, 184107.]; Tsirelson & Stash, 2020[Tsirelson, V. & Stash, A. (2020). Acta Cryst. B76, 769-778.]; Shteingolts et al., 2021[Shteingolts, S. A., Stash, A. I., Tsirelson, V. G. & Fayzullin, R. R. (2021). Chem. A Eur. J, 27, 7789-7809.]; Bartashevich et al., 2021[Bartashevich, E., Stash, A., Yushina, I., Minyaev, M., Bol'shakov, O., Rakitin, O. & Tsirelson, V. (2021). Acta Cryst. B77, 478-487.]). In this work, we aimed to analyze the properties inherent to each function of orbital-free DFT relevant to quantum crystallography. As far as it is possible, we followed general physical principles, regardless of any selected chemical compound, and only then we did consider the chemical bonds in a specific case.

Usually, a physicist explains the nature of properties of molecules and crystals by using potentials, forces and energy, whereas a chemist thinks in terms of a set of atoms and bonds between them (Tiana et al., 2011[Tiana, D., Francisco, E., Blanco, M. A., Macchi, P., Sironi, A. & Martín Pendás, A. (2011). Phys. Chem. Chem. Phys. 13, 5068-5077.]; Martin Pendás et al., 2012[Martın Pendas, A., Kohout, M., Blanco, M. A. & Francisco, E. (2012). In Modern Charge Density Analyses, edited by C. Gatti & P. Macchi, pp. 303-358. Springer, London.]; 2019[Martin Pendás, A., Casals-Sainz, J. L. & Francisco, E. (2019). Chem. Eur. J. 25, 309-314.]). As we demonstrated above, modern quantum crystallography, as an interdisciplinary science, is ready to combine both these approaches to obtain the deeply hidden electron-density and energy-related information about a crystal that was previously unavailable for X-ray diffraction analysis. In this article, we continued to develop this direction, following the approach proposed by Tsirelson & Stash (2020[Tsirelson, V. & Stash, A. (2020). Acta Cryst. B76, 769-778.]).

This work exploits the data of the accurate X-ray diffraction experiment. It is important that the electrostatic and exchange potentials in solids may be also directly measured by scanning transmission electron microscopy at subatomic spatial resolution and electron holography (Spence, 1993[Spence, J. C. H. (1993). Acta Cryst. A49, 231-260.]; Avilov et al., 1999[Avilov, A. S., Kuligin, A. K., Pietsch, U., Spence, J. C. H., Tsirelson, V. G. & Zuo, J. M. (1999). J. Appl. Cryst. 32, 1033-1038.]; Avilov & Tsirelson, 2001[Avilov, A. S. & Tsirelson, V. G. (2001). Crystallogr. Rep. 46, 556-571.]; Ogata et al., 2008[Ogata, Y., Tsuda, K. & Tanaka, M. (2008). Acta Cryst. A64, 587-597.]; Winkler et al., 2020[Winkler, F., Barthel, J., Dunin-Borkowski, R. E. & Müller-Caspary, K. (2020). Ultramicroscopy, 210, 112926.]). Therefore, static PAEM potential may be expressed via quantities, accessible for different experimental methods. Knowing PAEM, the kinetic potential is easily derived via equation (9)[link]. That is why it is very timely to analyze the foundations and drawbacks of this approach to make it affordable and clear to the broad crystallographic community.

Let us shortly summarize our results. There is no single, unambiguously defined scheme of energy decomposition (Andrés et al., 2019[Andrés, J., Ayers, P. W., Boto, R. A., Carbó-Dorca, R., Chermette, H., Cioslowski, J., Contreras-García, J., Cooper, D. L., Frenking, G., Gatti, C., Heidar-Zadeh, F., Joubert, L., Martín Pendás, A., Matito, E., Mayer, I., Misquitta, A. J., Mo, Y., Pilmé, J., Popelier, P. L. A., Rahm, M., Ramos-Cordoba, E., Salvador, P., Schwarz, W. H. E., Shahbazian, S., Silvi, B., Solà, M., Szalewicz, K., Tognetti, V., Weinhold, F. & Zins, É.-L. (2019). J. Comput. Chem. 40, 2248-2283.]). Therefore, we decomposed the total electronic energy into kinetic and potential (static) components, using clear physical and chemical arguments. Then, after some DFT-borrowed approximations, we moved on to the one-electron Euler equation linking the kinetic and static potentials, which are further decomposed into tractable components: the Pauli and Weizsacker potentials and the electrostatic and exchange potentials, respectively. The partial electron densities, associated with these potentials by the Poisson equation, were considered as well. All these functions were expressed through experimental electron density and its derivatives and applied to the study of chemical bonding in a heteromolecular crystal of ammonium hydro­oxalate oxalic acid dihydrate. Results were analyzed in terms of their physical content and limits of the applicability of a method. As a result, our approach allowed the decomposition of the total electron density into physically meaningful components and interpreting the information extracted from the precise X-ray diffraction experiment with a depth that exceeds that which is usually obtained in similar works. More precisely, we have developed two schemes for the decomposition of the total electron density corresponding to the previously proposed decomposition of the total energy of the crystal. One of them considers the partial charge densities associated with electrostatics, exchange and spin-independent wave properties of electrons, and the other one deals with their combinations. As a result, orbital-free quantum crystallography has received a new tool for bonding analysis.

We can outline the problems whose solution in the frame of orbital-free quantum crystallography is of interest in the near future. First of all, it is desirable to replace the imperfect Becke (1988[Becke, A. D. (1988). J. Chem. Phys. 88, 1053-1062.]) potential with a better one and to find the robust approximation for potential vx, resp. Second, it is interesting to consider the formation of a chemical bond as a result of the interference of electronic waves subject to Fermi–Dirac statistics and the spatial constraints dictated by a configuration of the attraction centers (i.e. by the nuclei disposition). And finally, the positions of minima in the Weizsäcker potential and partial density, separating K and L electron shells in any atom with Z > 2 in a molecule/crystal, represent the minimal space that the bound atom could occupy if it was in a hypothetical bosonic state. We found that the inner parts of electron density of bound atoms are slightly contracted, expanded, or almost invariable, depending on the bonding picture. Therefore, corresponding directional K-L `radii' may serve as intrinsic geometrical characteristics of the bound atoms in certain valence states. We plan to address these issues in our future work.

Supporting information


Funding information

The following funding is acknowledged: Russian Foundation of Basic Research (grant No. Project 19-03-00141a to Vladimir Tsirelson); Ministry of Science and Higher Education of the Russian Federation (contract to Adam Stash).

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