Matta, Boyd. The quantum theory of atoms in molecules

272 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

ec via rðr), ‘rðrÞ, and ‘2rðrÞ (reviewed elsewhere [38]). We consider here the gradient-corrected exchange energy density by Becke [89], which is expressed for spin-unpolarized (i.e. the closed-shell) systems as:

ex ¼ ex; uniform ð1=2Þ1=3	bX 2
	1 þ 6bX sinh 1ðXÞ rðrÞ4=3	ð9Þ

where:

e		3		3	1=3r4=3	r ; b		0:0042; X		‘rs	rÞj
	x; uniform ¼	4					¼		¼ j	ð4=3
			p			ð Þ
									rsðrÞ

ex; uniform is the Dirac–Slater exchange density for a uniform electron gas; the value of ð1=2ÞrðrÞ is assigned for the spin density rsðrÞ. We consider also the gradient-corrected correlation energy density ecðrÞ by Perdew–Burke–Ernzerhof [90]:

ecðrÞ ¼ ½ecðrÞ þ HðrÞ&rðrÞ

ð10Þ

where ecðrÞ is correlation energy per electron for a uniform electron gas [91] and:

ð Þ ¼

þ g

ð Þ 1 þ AðrÞt2ðrÞ þ A2ðrÞt4

ðrÞ

H r

g ln 1

b

t

2

r

1 þ AðrÞt2ðrÞ

;

AðrÞ ¼ g exp

ecðrÞ

1

; tðrÞ ¼ j‘rðrÞj=½2ksrðrÞ&;

ksðrÞ ¼ 2 p rðrÞ

;

b

3

1=6

g

(b ¼ 0:066725, g ¼ 0:031091). Approximations Eq. (9) and Eq. (10) are popular in DFT applications [38, 40, 88] and we use them to explicitly reveal regions of potential energy reduction in crystalline urea, CO(NH2)2, caused by exchange between electrons of the same spin and spin-independent electron correlation.

The structure of urea (space group P421m, Z ¼ 2 (2 mm)) is characterized by ribbons of hydrogen-bonded molecules arranged head-to-tail along the c axis (Fig. 10.7a). The plane of each ribbon is perpendicular to adjacent ribbons directed oppositely along the c axis. The oxygen atom of a carbonyl group is

_______________________________________________________________________________G

Fig. 10.7 Structure of crystalline urea (a) and distributions of the electronic energy density heðrÞ (b), the gradient-corrected exchange energy density ex (c), Dirac–Slater exchange energy density (d), gradient-corrected correlation energy density (e), and correlation energy density for a uniform electron gas (f ). Line intervals are Gð2; 4; 8Þ 10n a.u. ( 3 an a3); negative values are solid.

10.3 Approximate Electronic Energy Densities 273

274 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

H-bonded to the neighboring molecule in the same ribbon and, simultaneously, is involved in H-bonds with two adjusted ribbons. Distributions of the exchange and correlation energy densities in urea are shown in Fig. 10.7, with total electronic density heðrÞ Eq. (4). The energy density heðrÞ (Fig. 10.7b) achieves its lowest negative values on the intramolecular bond lines, revealing areas of concentration associated with bonding interactions and lone pairs. (These features cannot be attributed to the presence of the Laplacian term in Eq. (6) – the Hartree–Fock electronic energy density has the same features.) Simultaneously, alternating negative minima and positive maxima observed in the vicinity of nuclei have the shell structure of bonded O, N, and C atoms. At the same time, heðrÞ is slightly positive in the intermolecular hydrogen bonds in urea, where kinetic energy dominates. The function heðrÞ enables hydrogen bonds of di erent length to be distinguished: the longer H-bond (dO...H2 ¼ 2:064ð2Þ A˚ ) is characterizing by a minimum value of he; min ¼ þ0:00179 a.u. whereas the shorter H-bond (dO...H1 ¼ 2:007ð2Þ A˚ ) has the less positive value he; min ¼ þ0:00139 a.u. Similar behavior of heðrÞ in crystalline urea (148 K data) and a-oxalic acid dihydrate, C2H2O4 H2O, has been reported elsewhere [61, 92]. This also agrees with a general observation made for weakly bounded molecular systems [25] and weak and intermediate hydrogen bonds [93].

A map of the local exchange energy ex (Eq. 9), depicted in the same plane (Fig. 10.7c), in addition to deep energy wells in the vicinity of the nuclei, also has negative exchange-energy density bridges between bounded atoms, which contributes to the potential energy reduction during the crystal formation. The magnitudes of exchange contributions reflect the features of a bond, e.g. the shorter H-bond in urea is characterized by a lower value of ex ¼ 0:005 a.u. than its longer counterpart ( 0.004 a.u.).

Function ex does not reproduce explicitly such typical bonding features as the bond charge and electron lone-pair concentrations or core density alternations. Although the last of these may be made evident by calculating ‘2½ exðrÞ&, there is no need to do this because the Laplacian of electron density itself provides us with this information.

It is worth noting that the gradient-corrected exchange density Eq. (9) is quite similar to the Dirac–Slater exchange density for a uniform electron gas, ex; uniform, depicted in Fig. 10.7d. The main discrepancy is observed in the vicinity of nuclei whereas on the bond lines it is only 0.002 a.u.

A map of the local gradient-corrected correlation energy, ec, (Fig. 10.7e) shows the contribution of the electron correlation to the total electronic energy, which is independent of the electron spin. It also reveals the energy wells in the vicinity of the nuclei and the negative energy density bridges between bounded atoms, indicative of reduced potential energy. Simultaneously, in contrast with the function ex, the gradient-corrected correlation energy density ec has the typical bonding features of bond charge and lone-pair electron concentrations. These features are absent in the map of correlation energy for a uniform electron gas ec; uniform ¼ ecðrÞrðrÞ computed in accordance with Ref. [91] (Fig. 10.7f ). Similar results were

10.4 The Integrated Energy Quantities 275

obtained early for molecules [94]. Thus, analysis of the correlation energy density reconstructed from the experimental electron density in the gradient-corrected approximation can provide precise details of the bonding mechanisms.

10.4

The Integrated Energy Quantities

The average molecular energy calculated by use of the variational principle using the kinetic energy approximation, Eq. (6), is only qualitatively close to the experimental value [31, 60]. At the same time, computation of approximate gDFT , Eq. (6), with the Hartree–Fock wavefunctions leads to average kinetic energy di ering from the Hartree–Fock energy by approximately 1%. It is, therefore, worth determining the integrated energy-related quantities from the experimental ED combined with Eq. (6), especially because experimental r is close to the theoretical value.

According to Bader [1], the position space of a molecule or crystal may be divided into atomic basins separated by surfaces satisfying the zero-flux condition:

‘rðrÞ nðrÞ ¼ 0; Er A SiðrÞ

ð11Þ

These basins are identified with bonded atoms (pseudoatoms). An integral of any property, A(r), over the volume of an atom i, Wi:

ð

hAii ¼ AðrÞ dV ð12Þ

Wi

yields an average value of the property. These quantities are uniquely determined

Ð

because Wi ‘2rðrÞ dV ¼ 0. The sum of atomic contributions thus obtained yields the value of the property for a whole system and for the functional atomic groups, bonded molecules, and elementary cells in a crystal.

Atomic components of electronic energy, He, for H2O (in a-oxalic acid dihydrate, C2H2O4 H2O), NH3, and Cl2 molecules removed from a crystal are given

in Table 10.1. The integral				Wi ‘2rðrÞ dV over each of the atomic basins was less
than 10		3	a.u., thus the	latter number is an estimate of the integration error.
G				Ð

After summing, atomic contributions yield the electronic energy of a molecule. Quantum chemistry-calculated energy values He for free molecules (Table 10.1) are in reasonable agreement with ‘‘experimental’’ energies, despite the slight distortion of the experimental electron densities by the parent crystal environment. The largest discrepancy of 1.7% obtained for the Cl2 molecule may be attributed to the relatively low accuracy of the corresponding X-ray di raction experiment. We note, however, that even with the perfect X-ray di raction data energies of in-

27610 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

Table 10.1 Atomic contributions to the electronic energy, He, exchange energy, Ex, and correlation energy, Ec, calculated from the model electron densities for water (in a-oxalic acid dihydrate), ammonia, and chlorine molecules (all the molecules removed from a crystal). The total molecular values calculated by summing atomic contributions are also listed; the corresponding nonempirically calculated values are given in parentheses. All values are given in atomic units.

Molecule/atom	He	Ex	Ec
O	75.743	8.759	0.322
H2O H	0.298	0.162	0.011
H	0.296	0.160	0.011
Total	76.337	9.081	0.344
	( 76.171 [100])	( 8.946 [101])	( 0.371 [102])
NH3 N	55.300	7.129	0.290
H	0.301	0.171	0.012
Total	56.204	7.642	0.326
	( 56.326 [100])	( 7.670 [101])	( 0.340 [102])
Cl2 Cl	451.517	26.887	0.667
Total	903.034	53.774	1.334
	( 918.892 [100])	( 55.094 [101])	( 1.380 [102])

termolecular interaction or cohesion can hardly be obtained in this way, because their typical values are comparable with experimental uncertainty.

The atomic integrated values of the exchange energy, Ex, the correlation energy, Ec, and the corresponding molecular values are also given in Table 10.1. A maximum Ex discrepancy of 2.5% is observed for Cl2 molecule, whereas a maximum Ec discrepancy of 7.5% is found for H2O molecule. The last disagreement is partially the result of the di erent correlation energy definitions in quantum chemistry and DFT [88]. Overall agreement between experimentally derived and theoretically computed exchange and correlation energies is, nevertheless, reasonable. The same conclusion is valid for the crystalline wea (Table 10.2).

10.5

Concluding Remarks

Our approach consists in the joint use of the QTAMC and DFT to treat the electron density and density-dependent functions obtained with the multipole model is fitted to accurate X-ray di raction data. Despite current problems with adequate

10.5 Concluding Remarks 277

Table 10.2 Atomic volumes, Wi, restricted by zero-flux surfaces shown in the picture below, atomic electron populations, Pi, and

the integrated atomic electronic, He; i, exchange, Ex; i, and correlation, Ex; i, energies of the urea molecule computed by integration of corresponding densities derived from the X-ray synchrotron data at 123 K (see text). Only values for symmetry-independent atoms are given.

Atom	˚ 3	Pi, e	[a]	[a]	[a]
	Wi, A		He, i, a.u.	Ex, i, a.u.	Ec, i, a.u.
C	4.818	4.27(2)	37.108(1)	4.595(1)	0.143(1)
O	16.486	9.19(2)	75.498(1)	7.043(1)	0.324(1)
N	17.985	7.96(2)	55.502(1)	8.715(1)	0.281(1)
H1	3.847	0.65(2)	0.446(1)	0.237(1)	0.016(1)
H2	3.923	0.66(2)	0.450(1)	0.238(1)	0.016(1)

a The sum of the atomic energy contributions over a single molecule is225.402(1) a.u. The molecular Hartree–Fock/6-311G** energy is224.046 a.u., whereas the B3LYP/6-311G** method yields 225.401 a.u.

description of the electron density for shared atomic interactions (which can be overcome by use of a more sophisticated multipole model), the level of unification of the theoretical and experimental methods achieved substantially increases the amount of information derivable directly from X-ray di raction data. It reveals how the total electronic energy and its di erent components are distributed over a molecule and crystal and provides a real-space insight into the energetics of the chemical bond. We also hope that analysis of the exchange and correlation energy densities derived in di erent approximations from experimental electron density may be useful for improving existing DFT functionals.

It is worth mentioning that we did not consider all aspects of the aforesaid approach. Approximate determination of the electron localization function, localized-orbital locator, the local (and integrated) internal temperature of an electron gas, and associated entropy from the experimental electron density and its derivatives is reported elsewhere [92, 95, 96]; the kinetic energy density was described in these works by use of Eq. (6). Dirac–Slater exchange potential and one-electron potential determined using experimental electron density are de-

278 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

scribed elsewhere [27, 97]. Bonding analysis in terms of the critical points properties is also given elsewhere [18].

Calculation of all the energy characteristics from the electron density in this work was performed with the WinXPRO program [97, 98] using the relativistic many-configuration wavefunctions from Ref. [99].

Acknowledgments

I would like to express my deep gratitude to Richard Bader for longstanding scientific collaboration and personal friendship. The author thanks Dr A.I. Stash for collaboration and Professor W. Scherer and Dr H. Birkedal for making available the X-ray experimental data for a-oxalic acid dihydrate and urea. This work is supported by the Russian Ministry for Education (grant 2.1.1.5051) and the Russian Foundation for Basic Research (grant 04.03.33053).

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