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Atomic valence in molecular systems
Chemical Physics Letters 375 (2003) 45–53 www.elsevier.com/locate/cplett
Atomic valence in molecular systems R.C. Bochicchio
a,*
, L. Lain b, A. Torre
b
a
b
Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria 1428, Buenos Aires, Argentina Departamento de Quımica Fısica, Facultad de Ciencias, Universidad del Paıs Vasco, Apdo, 644 E-48080 Bilbao, Spain Received 16 April 2003; in final form 10 May 2003
Abstract Atomic valence in molecular systems is described as a partitioning of the hole distribution, the complementary part of the particle distribution. In this scheme, valence splits into three contributions, related to electron spin density, nonuniform occupancy of orbitals (nonpairing terms) and exchange density (pairing terms), respectively, and whose importance depends on the nature of the state of the system. Calculations carried out for correlated CI and Hartree– Fock state functions in both Mulliken and topological AIM type partitionings as well as theoretical results show the suitability of this formulation for describing valence concepts. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The effort dedicated to the determination and interpretation of the electron distribution by using the p-particle density matrices (p-RDM) has proven these matrices are invaluable tools in quantum theory of many-electron molecular systems [1]. However, no similar effort has been devoted to the study of the hole distribution, expressed by the p-hole reduced density matrices (p-HRDM), which are related to the p-RDM by means of the anticommutation rules of creation and annihilation fermion operators [2]. This stems from the fact that hole distribution is not so simple to relate to charge population as par-
*
Corresponding author. Fax: 54-11-45763357. E-mail address: [email protected] (R.C. Bochicchio).
ticle distribution does, mainly because its physical meaning is less evident. Thus, it is necessary to state clearly the features and applications of the hole distribution. The difference between the orbital highest population (i.e., two electrons) and the actual population, means how many electrons are needed to fill up the orbital. The orbital populations are described by the 1-particle reduced density matrix (1-RDM) while the lack of orbital population, or hole population, is described by the counterpart matrix, the 1-hole reduced density matrix (1-HRDM). On the other hand, valence is a classical quantity related to the capacity of bonding of an atom in a molecular system and is interpreted as the number of electrons needed to complete the outer shells following the octet rule [3]. Therefore, from the quantum mechanical point of view, these electrons can be regarded as filling the holes
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00805-4
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introduced above and thus classical valence is a good candidate to be described by means of hole distribution partitioning. The concept of valence has widely been used from the early days of chemical bonding theory [3]. The onset of modern quantum theory established the proper scenario to interpret such empirical concept as well as its close relation to the electron spin. The analogy between chemical structure and mathematical theory of invariants leads to WeylÕs formula from which one can calculate the valence of an isolated atom A as VA ¼ 2SðAÞ, where SðAÞ is the spin quantum number of the atom [4]. However, the atomic spin is not a good quantum number when the atom is immersed in a molecule and consequently WeylÕs formula cannot be used to estimate the capacity of chemical bonding of an atom in the molecular system. Several attempts to solve this problem led to empirical definitions that have been applied to closed and open shell systems at different levels of theory [5–12]. This Letter attempts to clarify the concept of atomic valence by providing a mathematical framework to carry out a partitioning of a general model of such a quantity into physical components. This formulation can be extended as to closed shell ground states as to ions, radicals or excited states, namely systems with a net unpaired number of electrons as well as hypervalent systems where the possibility of expanded valence shells must be taken into account [13]. The model is formulated in the second quantized language [14] and leads to a general definition for valence and its corresponding partitioning into terms related to spin density, orbital nonuniform occupancy and exchange density which describe different features of the electron distribution. Thus, the first two terms are of nonpairing nature and are related to the unpaired electron density through cumulant densities of the 2-particle reduced density matrix (2-RDM) as shown in [15,16] while the last contribution is of pairing nature leading to bonding concepts. Section 2 presents the theoretical derivation of these quantities along with a discussion of their physical meaning. In Section 3 the computational details and the numerical results are discussed. Finally, Section 4 is devoted to some additional remarks and conclusions.
2. Theory Let us consider a set fi; j; k; l; . . .g of orthonormal orbitals and the corresponding spin-orbitals fia ; ib ; ja ; jb ; k a ; k b ; la ; lb ; . . .g. Let us also consider the usual creation and annihilation fer mion operators cþ ir , cjr , respectively, obeying the anticommutation relation þ cþ ir cjr0 þ cjr0 cir ¼ dij drr0 ;
ð1Þ
where r ¼ a; b stands for the spin coordinates. Upon summation over spin coordinates the spin free version reads Eji þ Eij ¼ 2dij ;
ð2Þ cþ c i b jb
where Eji ¼ cþ stands for the first order i a c ja þ replacement operator [17]. The expectation values for the molecular state vector jLi, hLjEji jLi ¼ 1 i Dj are the spin-free one-particle reduced density a b matrix (1-RDM) elements (1 Dij ¼ 1 Dija þ 1 Dijb ). j þ þ Similarly, Ei ¼ cja cia þ cjb cib is the spin-free onehole replacement operator whose expectation i are the spin free reduced values hLjEji jLi ¼ 1 D j i one-hole density matrix elements (1-HRDM), 1 D j 1 i 1 ia 1 ib ð Dj ¼ Dja þ Djb Þ [2]. Then the particle number operator is X N^ ¼ dij Eji ; ð3Þ i;j
and the substitution of the Kronecker delta expressed by the average of Eq. (2) as dij ¼ 12ð1 Dij þ 1 i Dj Þ yields 1 X 1 i i 1 X 1 i i N^ ¼ Dj Ej ; D j Ej þ ð4Þ 2 i;j 2 i;j which under the averaging procedure leads to Þ: 2N ¼ trð1 D1 DÞ þ trð1 D1 D
ð5Þ
which represents the particle number conservation equation in its most general form within the molecular orbital theory and acts as the master equation for density partitioning. It is straightforward to note from Eq. (4) that the first term on the r.h.s. of Eq. (5) is the particle contribution to the conservation equation while the second term is associated to the hole one. This last term is that in which we are interested in and thus we will show how valence and free valence quantities arise from
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its partitioning. For this purpose, let us first consider the spin-free unpaired electron density u proposed by Takatsuka et al. [18,19] whose matrix form [15,20] reads X 1 i1 k uij ¼ 21 Dij Dk Dj ; ð6Þ k
The last term on the r.h.s. of Eq. (10) must be recognized as the ÔexchangeÕ contribution to the bond order or bonding population defined by [23,24] X 1 i1 j Dl Dk hijkiXA hjjliXB ; ð11Þ IXA XB ¼ ijkl
and its trace is (cf. Eq. (5)) Þ: trðuÞ ¼ 2N trð D DÞ ¼ trð D D 1
1
1
1
ð7Þ
Eq. (7) has been the departure point in [21] to define the Ômean number of holes in the systemÕ which has allowed us to derive the empirical formula and to establish the relations between ÔvalenceÕ and Ôfree valenceÕ within the formalism of the statistical population analysis. Now to continue our analysis we make explicit use of uij matrix elements that indicate the deviation from idempotency of the spin-free 1-RDM, 1 Dij . If the state function of the system is described by a Slater determinant with doubly occupied molecular orbitals (MO) all these matrix elements uij vanish. Thus the trace can be interpreted as the number of effectively unpaired electrons, Nu [18,19] P (cf. discussion in [21]), such that trðuÞ ¼ i uii ¼ Nu is the average deviation of orbital populations from the highest occupation number in the state of the N -electron system. The u matrix decomposition [18] in matrix form is 2
u ¼ ð1 Dspin Þ D;
ð8Þ
and therefore is straightforward to note that the remaining terms in Eq. (10) are the bonding capacity or valence of atom A X X 1 i 1 i1 j VXA ¼ 2 Dk hijkiXA Dl Dk hijkiXA hjjliXA : ik
where D ¼ Da þ Db collects the total deviation from the idempotency as contributions of the spin up (a) cloud and the spin down (b) cloud (cf. Appendix A). Eq. (7) can be written with the aid of its topological partitioning [15] as X XX uXA ¼ uik hijkiXA ; ð9Þ Nu ¼ XA
i;k
where hijkiXA means the molecular orbital overlap matrix over the atomic basins XA [22], and Eq. (6) leads to X X 1 i 1 i1 j uXA ¼ 2 Dk hijkiXA Dl Dk hijkiXA hjjliXA ik
X X
XB 6¼XA ijkl
ijkl 1
Dil 1 Djk hijkiXA hjjliXA ;
ð10Þ
ijkl
ð12Þ Thus Eq. (10) can bePwritten in a more compact form as uXA ¼ VXA XB 6¼XA IXA XB . An alternative partitioning of the N electrons of the system can be performed in Eq. (7) according to MullikenÕs criteria. In this scheme, as is well known, the partitioning is performed over the functions of the basis set instead of the real space so that the Mulliken version of Eqs. (11) and (12) are IAB ¼
A X B X i
1
Dij 1 Dji ;
j
and VA ¼
A X B XX B6¼A
XA
47
i
j
1
Dij 1 Dji þ
A X i
uii ¼
X
IAB þ uA ;
B6¼A
PA respectively, where the symbol i means that the sum is restricted to the orbitals centered on atom A. From now on the theoretical discussion will be done regarding the topological scheme but, as is easy to see, MullikenÕs description is similar and all the conclusions remain valid. Note that uA is the free valence FA of [21]. Eq. (12) fits well to the first empirical valence definition from a quantum mechanical point of view [5]. This definition is absolutely general within the nonrelativistic quantum theory, that is, it is valid for any molecular system and state function quality, i.e., noncorrelated or correlated state functions. The substitution of Eq. (8) into Eq. (12) leads to the valence partitioning as
48
VXA ¼ SXA DXA þ
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X
IXA XB ;
ð13Þ
XB 6¼XA
P where SXA ¼ ik ½ð1 Dspin Þ2 ik hijkiXA stands for the spin contribution to valence and DXA ¼ P density i ik Dk hijkiXA represents the degree of nonuniform occupancy of molecular orbital populations or lack of idempotency of the spin up and spin down 1-RDM. Therefore, we note that the atomic valence has contributions which take into account well defined features of the electron cloud distribution. Let us discuss each term in Eq. (13) from a physical point of view. The spin density contribution SXA for atom A is the weighted spin density with itself resulting in the square of such spin 1-RDM. It only contributes if the state spin is S 6¼ 0, otherwise 1 Dspin is a null matrix; thus nonpaired electrons on the system are responsible of such contribution. The other contribution DXA is (cf. Appendix A) the indicator of the lack of idempotency for both spin up and spin down electron one-particle density matrices which physically means that the electron population over each orbital is not uniform, namely, each orbital does not house the same number of particles. This feature is typical of correlated state functions; otherwise models fixing electron populations over molecular orbitals, i.e., independent particle models, assure idempotency for spin densities and thus no contribution comes from this term. Finally, the last contribution stands for the electron population from the exchange pair density matrix (ÔexchangeÕ part of the 2-RDM [25]) collecting all bonding situations with other atoms in the system. Some particular situations are of physical interest; for instance one can consider a closed shell molecular system described by a restricted Hartree–Fock (RHF) state function. In this case 1 Dspin is a null matrix and DXA 0 for all the atoms, because every molecular orbital houses only one spin up and one spin down electron and thus the atomic P valence has only exchange contributions VXA ¼ B6¼A IXA XB , which is the most known and classically considered formula for valence [5–7,11] as the sum of bond orders. If the electron configuration is consistent with singlet states described by correlated state P functions, the valence is VXA ¼ DXA þ B6¼A IXA XB because there are contributions of nonuniform-
population of the molecular orbitals but no spin contribution is present in such cases [21]. In other cases, i.e., multiplet states such as systems with net unpaired electrons like nonsinglet excited states, radicals and ions, all terms contribute to the calculation of VXA . The common feature of all these terms coming from the spin density and the nonuniformity matrix D is that both are terms that, in many body language, are of nonpairing nature i.e., terms that cannot couple two electron spins, while the ÔexchangeÕ part do.
3. Computations and results Numerical determinations have been carried out in order to test the above methodology to calculate atomic valences. For such a goal some simple selected molecular systems ranging from closed shell singlets to radicals, excited and hypervalent states have been chosen as a wide representative group which exhibits different bonding features to report atomic valence numbers and their contributions. Configuration interaction (CI) state functions in the single and double expansion approximation (SDCI) have been used for all systems and states. The calculations were performed using a GA U S S I A N 94 [26] code from which first-order reduced density matrices as well as the atomic overlap integrals hijjiXA were generated for singlet states while PR O A I M [27] was used to compute overlap integrals for all the others. In all cases, the employed basis sets have been 6-31G**. All molecular geometries were optimized for these basis sets within SDCI used scheme. The Mulliken type calculations have been carried out in the atomic orbital basis sets. As these basis sets are nonorthogonal, the matrix elements 1 Dij in the above equations must be replaced by ðPSÞij , where P and S stand for the usual charge density and ovelap matrices, respectively. The Hartree–Fock calculations share the same molecular geometries than those obtained from CI optimizations. In classical chemical theory, valence concept as well as bond orders are represented as integer numbers. Quantum physics has shown that the latter ones cannot completely be represented by such numbers in a lot of cases, i.e., no Lewis
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structures as those possessing bonds that involve more then two centers, and reach integer values in case of so called perfect covalent bonds. Therefore, the analysis of such deviations from the classical rule indicates ionic component of the bond, weak bonds or complex patterns as the above mentioned multicenter bonds. Tables 1 and 2 show topological (Bader type) and Mulliken schemes for partitioning the total number of electrons in the system for CI state functions. In these tables valence VXA ðVA Þ is reported as well as its contributions SXA ðSA Þ, DXA ðDA Þ and total bond orders (i.e., sum of bond orders for each atom), and number of unpaired electrons uA ðuXA Þ [15] (free valence FA or average number of holes in atom A in [21]). A detailed discussion over the differences between both schemes of partitioning can be found in [28] which may be useful for the present discussion. Table 3 collects Hartree–Fock results in both schemes of partitioning for the same systems.
49
Let us begin with the group of molecules with similar results in both schemes. These are H2 , N2 , HCl, NH3 , CH4 and C2 H4 . The H2 molecule presents similar values for each component in both schemes, which are close to those classically expected. The N2 system has identical values for the nonuniform component and thus the slight difference between the valence values (3.170 in the topological partitioning and 2.926 in the Mulliken one), comes from the bond order which is greater in the former partitioning. The same happens for HCl. Note that in both cases the nonuniformity of the electron cloud is important, 0.37. However, the bond orders are very close to integer numbers and consequently the bonds can be considered as covalent. A similar discussion is valid for NH3 and CH4 compounds. The valences are very close to integer numbers, except for Cl atom in the HCl system, which indicates an expansion of the valence shell from d functions localized on Cl atom according to well known results that these
Table 1 Topological partitioning: atomic valence, free valence and contributions System
Atom
H2 N2 LiH
H N Li H Cl H O H N H C H C H O H N H C H C H Cl F S F
HCl H2 O NH3 CH4 C2 H4 OH NH2 CH3 C2 H4 (Triplet) ClF3 SF6
SXA
0.196 0.010 0.173 0.016 0.135 0.024 0.170 0.030
DXA
P
VXA
uXA
)0.060 )0.346 )0.015 )0.186 )0.368 )0.058 )0.325 )0.033 )0.309 )0.044 )0.219 )0.060 )0.257 )0.055 )1.063 )0.033 )1.067 )0.046 )0.964 )0.068 )0.897 )0.081 )0.285 )0.253 )0.192 )0.083
0.955 2.822 0.196 0.196 0.994 0.994 1.300 0.658 2.579 0.895 3.775 1.065 3.789 1.061 0.657 0.657 1.737 0.888 2.876 1.029 3.014 1.013 2.703 1.081 2.766 1.072
1.015 3.170 0.211 0.382 1.362 1.052 1.626 0.690 2.887 0.939 3.995 1.125 4.045 1.115 1.916 0.700 2.977 0.950 3.969 1.121 4.081 1.123 2.988 1.343 2.849 1.265
0.060 0.346 0.015 0.186 0.368 0.058 0.325 0.033 0.309 0.044 0.219 0.060 0.257 0.055 1.259 0.043 1.240 0.062 1.098 0.091 1.067 0.110 0.285 0.253 0.083 0.192
BðB6¼AÞ IXA XB
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Table 2 Mulliken partitioning: atomic valence, free valence and contributions System
Atom
H2 N2 LiH
H N Li H Cl H O H N H C H C H O H N H C H C H Cl F S F
HCl H2 O NH3 CH4 C2 H4 OH NH2 CH3 C2 H4 (Triplet) ClF3 SF6
SA
0.190 0.015 0.169 0.018 0.142 0.021 0.177 0.026
DA
P
VA
uA
)0.060 )0.346 )0.075 )0.126 )0.366 )0.060 )0.302 )0.045 )0.298 )0.047 )0.250 )0.052 )0.275 )0.046 )1.058 )0.038 )1.080 )0.040 )1.060 )0.036 )0.974 )0.042 )0.294 )0.249 )0.178 )0.167
0.952 2.579 0.800 0.800 0.902 0.902 1.710 0.853 2.658 0.875 3.757 0.915 3.704 0.925 0.828 0.828 1.731 0.860 2.791 0.917 2.799 0.899 2.024 0.804 5.369 0.958
1.012 2.926 0.874 0.925 1.267 0.962 2.012 0.897 2.956 0.923 4.007 0.967 3.979 0.970 2.076 0.883 2.980 0.917 3.993 0.974 3.950 0.968 2.319 1.053 5.536 1.136
0.060 0.346 0.075 0.126 0.366 0.060 0.302 0.045 0.298 0.047 0.250 0.052 0.275 0.046 1.248 0.054 1.250 0.058 1.202 0.057 1.151 0.069 0.294 0.249 0.167 0.178
functions cannot be considered as polarization functions for such an atom [29]. This conclusion is also valid for the description at the Hartree–Fock level for both partitionings as shown in Table 3. Other molecules show large differences between both descriptions: LiH and H2 O. The LiH system shows a drastic difference; on one side the topological approach describes this molecule as an ionic Liþ H structure while Mulliken scheme, with a bonding population of 0.8, indicates a stronger covalent nature. Thus, the valence numbers are increased from 0.2 to 0.9. The lower value indicates that the 2s valence function of Li atom is hardly occupied and because the hole population is a weighted one, i.e., product with 1 D, provides such small number (cf. Eq. (8)). Not so drastic but important ionic component is observed in OH bonds in H2 O in the topological description. However Mulliken scheme describes this bond as possessing a more important covalent component. Both H2 O and LiH systems are classically con-
BðB6¼AÞ IAB
sidered as possessing typical ionic bonding nature; LiH of strong character while H2 O of moderate strength. The same features are also noted in Hartree–Fock treatment. The set of molecular systems with a net number of unpaired electrons, as molecular ions, radicals and some excited states, OH, NH2 , CH3 and first excited triplet state of C2 H4 , possess similar descriptions for both partitioning schemes. The common feature of all of them is that their valence numbers are very close to those expected from classical chemistry. Only H atom in OH system has a value which is lower than one due to the ionic nature of this bond. This shows that the electron is in the deeper region of the 1s cloud and thus its capacity of bonding becomes lower in the topological description than in the classical one. The total free valence, i.e., the number of unpaired electrons or average number of holes of heavy atoms, is as expected very close to one in OH, NH2 and CH3 and C2 H4 triplet states.
R.C. Bochicchio et al. / Chemical Physics Letters 375 (2003) 45–53
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Table 3 Hartree–Fock atomic valence: Mulliken and topological partitionings System
Atom
VA
H2 N2 LiH
H N Li H Cl H O H N H C H C H O H N H C H C H Cl F S F
1.000 2.802 0.824 0.824 0.946 0.946 1.764 0.880 2.753 0.908 3.915 0.953 3.892 0.958 1.854 0.863 2.798 0.901 3.900 0.960 3.865 0.956 2.029 0.786 5.341 0.947
HCl H2 O NH3 CH4 C2 H4 OH NH2 CH3 C2 H4 (Triplet) ClF3 SF6
uA
0.996 0.004 0.992 0.004 0.991 0.003 0.964 0.018
The remaining systems to be discussed are considered as hypervalent molecules [13] because of the expansion of the outer or valence shell electrons of some atoms. Hence, higher angular momentum atomic shells and lone pairs participate in bondings. Therefore, coordination numbers are increased so that the valence numbers become higher than those found for the same atoms in normal systems. Let us analyze these cases and compare the results from both schemes. Chlorine atom in ClF3 compound shows a valence close to 3 in the topological scheme and as is obvious the coordination number is the same. This number is composed of 2.7 total bonding population and )0.3 nonuniform electron contribution. Thus, the number of unpaired electrons is 0.3. A high valence is also present in fluor atoms. Mulliken partitioning describes this situation in a very different manner. Cl valence is 2.3, appreciably lower than in the former partitioning because the
VXA 1.000 3.036 0.194 0.194 1.022 1.022 1.265 0.639 2.627 0.909 3.921 1.117 3.982 0.958 1.630 0.639 2.746 0.915 3.897 1.113 4.020 1.117 2.719 1.081 2.645 1.057
uXA
Bond
IAB
IXA XB
OH HH0 NH HH0 CH HH0 CC CH OH
1.000 2.802 0.824 0.824 0.946 0.946 0.882 0.000 0.918 0.000 0.978 0.000 1.968 0.975 0.858
1.000 3.036 0.194 0.194 1.022 1.022 0.633 0.639 0.876 0.012 0.219 0.045 1.887 0.982 0.635
NH HH0 CH HH0 CC CH ClF FF0 SF FF0
0.903 0.000 0.970 0.000 1.010 0.962 0.676 0.055 0.890 0.014
0.885 0.018 0.999 0.040 1.096 0.978 0.906 0.087 0.441 0.152
LiH HCl
0.996 0.004 0.976 0.012 0.899 0.034 0.888 0.056
bonding is also lower. However, free valence remains unchanged. The same happens for all F atoms. SF6 system possesses valence numbers appreciably lower for the topological scheme than for the Mulliken description in S atom. For this atom, these values are close to 3 in topological scheme and close to 6 in Mulliken one as the coordination number indicates. This result can be regarded as an example of the fact that coordination numbers and valence indices are not related in a simple manner. The coordination number gives a graphic idea to identify the number of bonds present around an atom while the valence index is an electronic magnitude which indicates the electron vacant sites that an atom can fill in the molecular system. F atoms have valence numbers slightly higher than 1 for both partitionings, which indicates that F atoms also possess an outer shell expansion. Free valences are low for all atoms.
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4. Final remarks and conclusions The valence concepts described in Section 1 are associated to hole distributions and arise from the fundamental expressions in fermion algebra, i.e., the anticommutation creation/annihilation relationships. In this sense, the hole distribution can be considered as the counterpart of the particle distribution and thus it has an equivalent hierarchical value and allows a suitable partitioning. The present model to describe and to determine atomic valences in molecular systems is completely valid within the nonrelativistic quantum physics and so it is suitable for any quality of state function used to describe the system as can be seen from its derivation. Moreover, the set of systems considered to test this formulation is wide enough to include different bonding situations: radicals, excited and hypervalent molecules, the numerical calculations fit well to the expected values for most of them and also show the expected differences between the reported partitioning schemes. It leads to conclude that although valence has been classically defined for closed shell molecules in its ground state, the model proposed in this Letter can be extended to other situations in which classical theory does not provide a definite answer such as excited and hypervalent systems. Extensions in this direction will be reported in the future [30]. At this moment it is worthwhile to mention a striking and attractive new line of research recently developed which is a direct link between quantum chemistry and condensed matter physics [31–34]. These works share the quantities reported in the present Letter and constitute a natural field of application to more complex systems. Research in this direction is currently carried out in our laboratories and will be reported elsewhere [35].
Acknowledgements R.C.B. acknowledges grants in aid from the University of Buenos Aires (project No. X-119) and Consejo Nacional de Investigaciones Cientıficas y Tecnicas, Rep ublica Argentina (PIP No. 02151/01). L.L. and A.T. thank DGI (Spain) and the Universidad del Pais Vasco for their sup-
port with the projects No. BQU 2000-0216 and 00039.310-EB 7730/2000, respectively.
Appendix A. u matrix structure The u matrix can be split into two contributions, one related to the spin density of the electron cloud and the other one to the nonuniform occupancy of the molecular orbitals. In matrix notation u reads u ¼ 21 D 1 D 2 :
ðA:1Þ
The formulation of the one-particle reduced density matrix as the sum of two spin blocks which correspond to spin up (a) and spin down (b), respectively, 1 D ¼ 1 Da þ 1 Db , leads to 2
u ¼ 21 Da þ 21 Db ð1 Da þ 1 Db Þ :
ðA:2Þ
The nonuniformity of each spin block of the oneparticle reduced density matrix, i.e., its lack of 2 idempotency, is formulated as 1 Dr ¼ 1 Dr þ 12Dr , r where r ¼ a or b and D matrix stands for the deviation of idempotency of the Hartree–Fock matrices. The substitution of this formula in Eq. (A.2) yields u ¼ ð1 Dspin Þ2 D; 1
spin
1
ðA:3Þ a
1
b
where D ¼ D D is the spin reduced density matrix and D ¼ Da þ Db collects the total deviation from idempotency as contributions of the spin up (a) and the spin down (b) clouds, respectively.
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[21] R.C. Bochicchio, J. Mol. Struct. (Theochem) 429 (1998) 229. [22] R.F.W. Bader, Atoms in Molecules. A Quantum Theory, Clarendon Press, Oxford, 1994, and references cited therein. [23] R.C. Bochicchio, L. Lain, A. Torre, R. Ponec, J. Math. Chem. 28 (2000) 83. [24] A. Torre, L. Lain, R.C. Bochicchio, R. Ponec, J. Math. Chem. 32 (2002) 241. [25] L. Lain, A. Torre, R.C. Bochicchio, J. Chem. Phys. 117 (2002) 5497. [26] M.J. Frisch et al., GA U S S I A N 94, Revision D.4, Gaussian Inc, Pittsburgh, PA, 1995. [27] F.W. Biegler-K€ oning, R.F.W. Bader, T.H. Tang, J. Comp. Chem. 3 (1982) 317. [28] R. Ponec, J. Roithova, X. Girones, L. Lain, A. Torre, R.C. Bochicchio, J. Phys. Chem. A 6 (2002) 1019. [29] I. Mayer, P.R. Surjan, Acta Chim. Hung. 117 (1984) 85. [30] R.C. Bochicchio, L. Lain, A. Torre, submitted. [31] A.E. Clark, E.R. Davidson, J. Chem. Phys. 115 (2001) 7382. [32] E.R. Davidson, A.E. Clark, Mol. Phys. 100 (2002) 373. [33] E.R. Davidson, A.E. Clark, J. Phys. Chem. A 106 (2002) 7456. [34] A.E. Clark, E.R. Davidson, Int. J. Quant. Chem. 93 (2003) 384. [35] R.C. Bochicchio, L. Lain, A. Torre, in preparation.
Atomic valence in molecular systems R.C. Bochicchio
a,*
, L. Lain b, A. Torre
b
a
b
Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria 1428, Buenos Aires, Argentina Departamento de Quımica Fısica, Facultad de Ciencias, Universidad del Paıs Vasco, Apdo, 644 E-48080 Bilbao, Spain Received 16 April 2003; in final form 10 May 2003
Abstract Atomic valence in molecular systems is described as a partitioning of the hole distribution, the complementary part of the particle distribution. In this scheme, valence splits into three contributions, related to electron spin density, nonuniform occupancy of orbitals (nonpairing terms) and exchange density (pairing terms), respectively, and whose importance depends on the nature of the state of the system. Calculations carried out for correlated CI and Hartree– Fock state functions in both Mulliken and topological AIM type partitionings as well as theoretical results show the suitability of this formulation for describing valence concepts. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The effort dedicated to the determination and interpretation of the electron distribution by using the p-particle density matrices (p-RDM) has proven these matrices are invaluable tools in quantum theory of many-electron molecular systems [1]. However, no similar effort has been devoted to the study of the hole distribution, expressed by the p-hole reduced density matrices (p-HRDM), which are related to the p-RDM by means of the anticommutation rules of creation and annihilation fermion operators [2]. This stems from the fact that hole distribution is not so simple to relate to charge population as par-
*
Corresponding author. Fax: 54-11-45763357. E-mail address: [email protected] (R.C. Bochicchio).
ticle distribution does, mainly because its physical meaning is less evident. Thus, it is necessary to state clearly the features and applications of the hole distribution. The difference between the orbital highest population (i.e., two electrons) and the actual population, means how many electrons are needed to fill up the orbital. The orbital populations are described by the 1-particle reduced density matrix (1-RDM) while the lack of orbital population, or hole population, is described by the counterpart matrix, the 1-hole reduced density matrix (1-HRDM). On the other hand, valence is a classical quantity related to the capacity of bonding of an atom in a molecular system and is interpreted as the number of electrons needed to complete the outer shells following the octet rule [3]. Therefore, from the quantum mechanical point of view, these electrons can be regarded as filling the holes
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00805-4
46
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introduced above and thus classical valence is a good candidate to be described by means of hole distribution partitioning. The concept of valence has widely been used from the early days of chemical bonding theory [3]. The onset of modern quantum theory established the proper scenario to interpret such empirical concept as well as its close relation to the electron spin. The analogy between chemical structure and mathematical theory of invariants leads to WeylÕs formula from which one can calculate the valence of an isolated atom A as VA ¼ 2SðAÞ, where SðAÞ is the spin quantum number of the atom [4]. However, the atomic spin is not a good quantum number when the atom is immersed in a molecule and consequently WeylÕs formula cannot be used to estimate the capacity of chemical bonding of an atom in the molecular system. Several attempts to solve this problem led to empirical definitions that have been applied to closed and open shell systems at different levels of theory [5–12]. This Letter attempts to clarify the concept of atomic valence by providing a mathematical framework to carry out a partitioning of a general model of such a quantity into physical components. This formulation can be extended as to closed shell ground states as to ions, radicals or excited states, namely systems with a net unpaired number of electrons as well as hypervalent systems where the possibility of expanded valence shells must be taken into account [13]. The model is formulated in the second quantized language [14] and leads to a general definition for valence and its corresponding partitioning into terms related to spin density, orbital nonuniform occupancy and exchange density which describe different features of the electron distribution. Thus, the first two terms are of nonpairing nature and are related to the unpaired electron density through cumulant densities of the 2-particle reduced density matrix (2-RDM) as shown in [15,16] while the last contribution is of pairing nature leading to bonding concepts. Section 2 presents the theoretical derivation of these quantities along with a discussion of their physical meaning. In Section 3 the computational details and the numerical results are discussed. Finally, Section 4 is devoted to some additional remarks and conclusions.
2. Theory Let us consider a set fi; j; k; l; . . .g of orthonormal orbitals and the corresponding spin-orbitals fia ; ib ; ja ; jb ; k a ; k b ; la ; lb ; . . .g. Let us also consider the usual creation and annihilation fer mion operators cþ ir , cjr , respectively, obeying the anticommutation relation þ cþ ir cjr0 þ cjr0 cir ¼ dij drr0 ;
ð1Þ
where r ¼ a; b stands for the spin coordinates. Upon summation over spin coordinates the spin free version reads Eji þ Eij ¼ 2dij ;
ð2Þ cþ c i b jb
where Eji ¼ cþ stands for the first order i a c ja þ replacement operator [17]. The expectation values for the molecular state vector jLi, hLjEji jLi ¼ 1 i Dj are the spin-free one-particle reduced density a b matrix (1-RDM) elements (1 Dij ¼ 1 Dija þ 1 Dijb ). j þ þ Similarly, Ei ¼ cja cia þ cjb cib is the spin-free onehole replacement operator whose expectation i are the spin free reduced values hLjEji jLi ¼ 1 D j i one-hole density matrix elements (1-HRDM), 1 D j 1 i 1 ia 1 ib ð Dj ¼ Dja þ Djb Þ [2]. Then the particle number operator is X N^ ¼ dij Eji ; ð3Þ i;j
and the substitution of the Kronecker delta expressed by the average of Eq. (2) as dij ¼ 12ð1 Dij þ 1 i Dj Þ yields 1 X 1 i i 1 X 1 i i N^ ¼ Dj Ej ; D j Ej þ ð4Þ 2 i;j 2 i;j which under the averaging procedure leads to Þ: 2N ¼ trð1 D1 DÞ þ trð1 D1 D
ð5Þ
which represents the particle number conservation equation in its most general form within the molecular orbital theory and acts as the master equation for density partitioning. It is straightforward to note from Eq. (4) that the first term on the r.h.s. of Eq. (5) is the particle contribution to the conservation equation while the second term is associated to the hole one. This last term is that in which we are interested in and thus we will show how valence and free valence quantities arise from
R.C. Bochicchio et al. / Chemical Physics Letters 375 (2003) 45–53
its partitioning. For this purpose, let us first consider the spin-free unpaired electron density u proposed by Takatsuka et al. [18,19] whose matrix form [15,20] reads X 1 i1 k uij ¼ 21 Dij Dk Dj ; ð6Þ k
The last term on the r.h.s. of Eq. (10) must be recognized as the ÔexchangeÕ contribution to the bond order or bonding population defined by [23,24] X 1 i1 j Dl Dk hijkiXA hjjliXB ; ð11Þ IXA XB ¼ ijkl
and its trace is (cf. Eq. (5)) Þ: trðuÞ ¼ 2N trð D DÞ ¼ trð D D 1
1
1
1
ð7Þ
Eq. (7) has been the departure point in [21] to define the Ômean number of holes in the systemÕ which has allowed us to derive the empirical formula and to establish the relations between ÔvalenceÕ and Ôfree valenceÕ within the formalism of the statistical population analysis. Now to continue our analysis we make explicit use of uij matrix elements that indicate the deviation from idempotency of the spin-free 1-RDM, 1 Dij . If the state function of the system is described by a Slater determinant with doubly occupied molecular orbitals (MO) all these matrix elements uij vanish. Thus the trace can be interpreted as the number of effectively unpaired electrons, Nu [18,19] P (cf. discussion in [21]), such that trðuÞ ¼ i uii ¼ Nu is the average deviation of orbital populations from the highest occupation number in the state of the N -electron system. The u matrix decomposition [18] in matrix form is 2
u ¼ ð1 Dspin Þ D;
ð8Þ
and therefore is straightforward to note that the remaining terms in Eq. (10) are the bonding capacity or valence of atom A X X 1 i 1 i1 j VXA ¼ 2 Dk hijkiXA Dl Dk hijkiXA hjjliXA : ik
where D ¼ Da þ Db collects the total deviation from the idempotency as contributions of the spin up (a) cloud and the spin down (b) cloud (cf. Appendix A). Eq. (7) can be written with the aid of its topological partitioning [15] as X XX uXA ¼ uik hijkiXA ; ð9Þ Nu ¼ XA
i;k
where hijkiXA means the molecular orbital overlap matrix over the atomic basins XA [22], and Eq. (6) leads to X X 1 i 1 i1 j uXA ¼ 2 Dk hijkiXA Dl Dk hijkiXA hjjliXA ik
X X
XB 6¼XA ijkl
ijkl 1
Dil 1 Djk hijkiXA hjjliXA ;
ð10Þ
ijkl
ð12Þ Thus Eq. (10) can bePwritten in a more compact form as uXA ¼ VXA XB 6¼XA IXA XB . An alternative partitioning of the N electrons of the system can be performed in Eq. (7) according to MullikenÕs criteria. In this scheme, as is well known, the partitioning is performed over the functions of the basis set instead of the real space so that the Mulliken version of Eqs. (11) and (12) are IAB ¼
A X B X i
1
Dij 1 Dji ;
j
and VA ¼
A X B XX B6¼A
XA
47
i
j
1
Dij 1 Dji þ
A X i
uii ¼
X
IAB þ uA ;
B6¼A
PA respectively, where the symbol i means that the sum is restricted to the orbitals centered on atom A. From now on the theoretical discussion will be done regarding the topological scheme but, as is easy to see, MullikenÕs description is similar and all the conclusions remain valid. Note that uA is the free valence FA of [21]. Eq. (12) fits well to the first empirical valence definition from a quantum mechanical point of view [5]. This definition is absolutely general within the nonrelativistic quantum theory, that is, it is valid for any molecular system and state function quality, i.e., noncorrelated or correlated state functions. The substitution of Eq. (8) into Eq. (12) leads to the valence partitioning as
48
VXA ¼ SXA DXA þ
R.C. Bochicchio et al. / Chemical Physics Letters 375 (2003) 45–53
X
IXA XB ;
ð13Þ
XB 6¼XA
P where SXA ¼ ik ½ð1 Dspin Þ2 ik hijkiXA stands for the spin contribution to valence and DXA ¼ P density i ik Dk hijkiXA represents the degree of nonuniform occupancy of molecular orbital populations or lack of idempotency of the spin up and spin down 1-RDM. Therefore, we note that the atomic valence has contributions which take into account well defined features of the electron cloud distribution. Let us discuss each term in Eq. (13) from a physical point of view. The spin density contribution SXA for atom A is the weighted spin density with itself resulting in the square of such spin 1-RDM. It only contributes if the state spin is S 6¼ 0, otherwise 1 Dspin is a null matrix; thus nonpaired electrons on the system are responsible of such contribution. The other contribution DXA is (cf. Appendix A) the indicator of the lack of idempotency for both spin up and spin down electron one-particle density matrices which physically means that the electron population over each orbital is not uniform, namely, each orbital does not house the same number of particles. This feature is typical of correlated state functions; otherwise models fixing electron populations over molecular orbitals, i.e., independent particle models, assure idempotency for spin densities and thus no contribution comes from this term. Finally, the last contribution stands for the electron population from the exchange pair density matrix (ÔexchangeÕ part of the 2-RDM [25]) collecting all bonding situations with other atoms in the system. Some particular situations are of physical interest; for instance one can consider a closed shell molecular system described by a restricted Hartree–Fock (RHF) state function. In this case 1 Dspin is a null matrix and DXA 0 for all the atoms, because every molecular orbital houses only one spin up and one spin down electron and thus the atomic P valence has only exchange contributions VXA ¼ B6¼A IXA XB , which is the most known and classically considered formula for valence [5–7,11] as the sum of bond orders. If the electron configuration is consistent with singlet states described by correlated state P functions, the valence is VXA ¼ DXA þ B6¼A IXA XB because there are contributions of nonuniform-
population of the molecular orbitals but no spin contribution is present in such cases [21]. In other cases, i.e., multiplet states such as systems with net unpaired electrons like nonsinglet excited states, radicals and ions, all terms contribute to the calculation of VXA . The common feature of all these terms coming from the spin density and the nonuniformity matrix D is that both are terms that, in many body language, are of nonpairing nature i.e., terms that cannot couple two electron spins, while the ÔexchangeÕ part do.
3. Computations and results Numerical determinations have been carried out in order to test the above methodology to calculate atomic valences. For such a goal some simple selected molecular systems ranging from closed shell singlets to radicals, excited and hypervalent states have been chosen as a wide representative group which exhibits different bonding features to report atomic valence numbers and their contributions. Configuration interaction (CI) state functions in the single and double expansion approximation (SDCI) have been used for all systems and states. The calculations were performed using a GA U S S I A N 94 [26] code from which first-order reduced density matrices as well as the atomic overlap integrals hijjiXA were generated for singlet states while PR O A I M [27] was used to compute overlap integrals for all the others. In all cases, the employed basis sets have been 6-31G**. All molecular geometries were optimized for these basis sets within SDCI used scheme. The Mulliken type calculations have been carried out in the atomic orbital basis sets. As these basis sets are nonorthogonal, the matrix elements 1 Dij in the above equations must be replaced by ðPSÞij , where P and S stand for the usual charge density and ovelap matrices, respectively. The Hartree–Fock calculations share the same molecular geometries than those obtained from CI optimizations. In classical chemical theory, valence concept as well as bond orders are represented as integer numbers. Quantum physics has shown that the latter ones cannot completely be represented by such numbers in a lot of cases, i.e., no Lewis
R.C. Bochicchio et al. / Chemical Physics Letters 375 (2003) 45–53
structures as those possessing bonds that involve more then two centers, and reach integer values in case of so called perfect covalent bonds. Therefore, the analysis of such deviations from the classical rule indicates ionic component of the bond, weak bonds or complex patterns as the above mentioned multicenter bonds. Tables 1 and 2 show topological (Bader type) and Mulliken schemes for partitioning the total number of electrons in the system for CI state functions. In these tables valence VXA ðVA Þ is reported as well as its contributions SXA ðSA Þ, DXA ðDA Þ and total bond orders (i.e., sum of bond orders for each atom), and number of unpaired electrons uA ðuXA Þ [15] (free valence FA or average number of holes in atom A in [21]). A detailed discussion over the differences between both schemes of partitioning can be found in [28] which may be useful for the present discussion. Table 3 collects Hartree–Fock results in both schemes of partitioning for the same systems.
49
Let us begin with the group of molecules with similar results in both schemes. These are H2 , N2 , HCl, NH3 , CH4 and C2 H4 . The H2 molecule presents similar values for each component in both schemes, which are close to those classically expected. The N2 system has identical values for the nonuniform component and thus the slight difference between the valence values (3.170 in the topological partitioning and 2.926 in the Mulliken one), comes from the bond order which is greater in the former partitioning. The same happens for HCl. Note that in both cases the nonuniformity of the electron cloud is important, 0.37. However, the bond orders are very close to integer numbers and consequently the bonds can be considered as covalent. A similar discussion is valid for NH3 and CH4 compounds. The valences are very close to integer numbers, except for Cl atom in the HCl system, which indicates an expansion of the valence shell from d functions localized on Cl atom according to well known results that these
Table 1 Topological partitioning: atomic valence, free valence and contributions System
Atom
H2 N2 LiH
H N Li H Cl H O H N H C H C H O H N H C H C H Cl F S F
HCl H2 O NH3 CH4 C2 H4 OH NH2 CH3 C2 H4 (Triplet) ClF3 SF6
SXA
0.196 0.010 0.173 0.016 0.135 0.024 0.170 0.030
DXA
P
VXA
uXA
)0.060 )0.346 )0.015 )0.186 )0.368 )0.058 )0.325 )0.033 )0.309 )0.044 )0.219 )0.060 )0.257 )0.055 )1.063 )0.033 )1.067 )0.046 )0.964 )0.068 )0.897 )0.081 )0.285 )0.253 )0.192 )0.083
0.955 2.822 0.196 0.196 0.994 0.994 1.300 0.658 2.579 0.895 3.775 1.065 3.789 1.061 0.657 0.657 1.737 0.888 2.876 1.029 3.014 1.013 2.703 1.081 2.766 1.072
1.015 3.170 0.211 0.382 1.362 1.052 1.626 0.690 2.887 0.939 3.995 1.125 4.045 1.115 1.916 0.700 2.977 0.950 3.969 1.121 4.081 1.123 2.988 1.343 2.849 1.265
0.060 0.346 0.015 0.186 0.368 0.058 0.325 0.033 0.309 0.044 0.219 0.060 0.257 0.055 1.259 0.043 1.240 0.062 1.098 0.091 1.067 0.110 0.285 0.253 0.083 0.192
BðB6¼AÞ IXA XB
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Table 2 Mulliken partitioning: atomic valence, free valence and contributions System
Atom
H2 N2 LiH
H N Li H Cl H O H N H C H C H O H N H C H C H Cl F S F
HCl H2 O NH3 CH4 C2 H4 OH NH2 CH3 C2 H4 (Triplet) ClF3 SF6
SA
0.190 0.015 0.169 0.018 0.142 0.021 0.177 0.026
DA
P
VA
uA
)0.060 )0.346 )0.075 )0.126 )0.366 )0.060 )0.302 )0.045 )0.298 )0.047 )0.250 )0.052 )0.275 )0.046 )1.058 )0.038 )1.080 )0.040 )1.060 )0.036 )0.974 )0.042 )0.294 )0.249 )0.178 )0.167
0.952 2.579 0.800 0.800 0.902 0.902 1.710 0.853 2.658 0.875 3.757 0.915 3.704 0.925 0.828 0.828 1.731 0.860 2.791 0.917 2.799 0.899 2.024 0.804 5.369 0.958
1.012 2.926 0.874 0.925 1.267 0.962 2.012 0.897 2.956 0.923 4.007 0.967 3.979 0.970 2.076 0.883 2.980 0.917 3.993 0.974 3.950 0.968 2.319 1.053 5.536 1.136
0.060 0.346 0.075 0.126 0.366 0.060 0.302 0.045 0.298 0.047 0.250 0.052 0.275 0.046 1.248 0.054 1.250 0.058 1.202 0.057 1.151 0.069 0.294 0.249 0.167 0.178
functions cannot be considered as polarization functions for such an atom [29]. This conclusion is also valid for the description at the Hartree–Fock level for both partitionings as shown in Table 3. Other molecules show large differences between both descriptions: LiH and H2 O. The LiH system shows a drastic difference; on one side the topological approach describes this molecule as an ionic Liþ H structure while Mulliken scheme, with a bonding population of 0.8, indicates a stronger covalent nature. Thus, the valence numbers are increased from 0.2 to 0.9. The lower value indicates that the 2s valence function of Li atom is hardly occupied and because the hole population is a weighted one, i.e., product with 1 D, provides such small number (cf. Eq. (8)). Not so drastic but important ionic component is observed in OH bonds in H2 O in the topological description. However Mulliken scheme describes this bond as possessing a more important covalent component. Both H2 O and LiH systems are classically con-
BðB6¼AÞ IAB
sidered as possessing typical ionic bonding nature; LiH of strong character while H2 O of moderate strength. The same features are also noted in Hartree–Fock treatment. The set of molecular systems with a net number of unpaired electrons, as molecular ions, radicals and some excited states, OH, NH2 , CH3 and first excited triplet state of C2 H4 , possess similar descriptions for both partitioning schemes. The common feature of all of them is that their valence numbers are very close to those expected from classical chemistry. Only H atom in OH system has a value which is lower than one due to the ionic nature of this bond. This shows that the electron is in the deeper region of the 1s cloud and thus its capacity of bonding becomes lower in the topological description than in the classical one. The total free valence, i.e., the number of unpaired electrons or average number of holes of heavy atoms, is as expected very close to one in OH, NH2 and CH3 and C2 H4 triplet states.
R.C. Bochicchio et al. / Chemical Physics Letters 375 (2003) 45–53
51
Table 3 Hartree–Fock atomic valence: Mulliken and topological partitionings System
Atom
VA
H2 N2 LiH
H N Li H Cl H O H N H C H C H O H N H C H C H Cl F S F
1.000 2.802 0.824 0.824 0.946 0.946 1.764 0.880 2.753 0.908 3.915 0.953 3.892 0.958 1.854 0.863 2.798 0.901 3.900 0.960 3.865 0.956 2.029 0.786 5.341 0.947
HCl H2 O NH3 CH4 C2 H4 OH NH2 CH3 C2 H4 (Triplet) ClF3 SF6
uA
0.996 0.004 0.992 0.004 0.991 0.003 0.964 0.018
The remaining systems to be discussed are considered as hypervalent molecules [13] because of the expansion of the outer or valence shell electrons of some atoms. Hence, higher angular momentum atomic shells and lone pairs participate in bondings. Therefore, coordination numbers are increased so that the valence numbers become higher than those found for the same atoms in normal systems. Let us analyze these cases and compare the results from both schemes. Chlorine atom in ClF3 compound shows a valence close to 3 in the topological scheme and as is obvious the coordination number is the same. This number is composed of 2.7 total bonding population and )0.3 nonuniform electron contribution. Thus, the number of unpaired electrons is 0.3. A high valence is also present in fluor atoms. Mulliken partitioning describes this situation in a very different manner. Cl valence is 2.3, appreciably lower than in the former partitioning because the
VXA 1.000 3.036 0.194 0.194 1.022 1.022 1.265 0.639 2.627 0.909 3.921 1.117 3.982 0.958 1.630 0.639 2.746 0.915 3.897 1.113 4.020 1.117 2.719 1.081 2.645 1.057
uXA
Bond
IAB
IXA XB
OH HH0 NH HH0 CH HH0 CC CH OH
1.000 2.802 0.824 0.824 0.946 0.946 0.882 0.000 0.918 0.000 0.978 0.000 1.968 0.975 0.858
1.000 3.036 0.194 0.194 1.022 1.022 0.633 0.639 0.876 0.012 0.219 0.045 1.887 0.982 0.635
NH HH0 CH HH0 CC CH ClF FF0 SF FF0
0.903 0.000 0.970 0.000 1.010 0.962 0.676 0.055 0.890 0.014
0.885 0.018 0.999 0.040 1.096 0.978 0.906 0.087 0.441 0.152
LiH HCl
0.996 0.004 0.976 0.012 0.899 0.034 0.888 0.056
bonding is also lower. However, free valence remains unchanged. The same happens for all F atoms. SF6 system possesses valence numbers appreciably lower for the topological scheme than for the Mulliken description in S atom. For this atom, these values are close to 3 in topological scheme and close to 6 in Mulliken one as the coordination number indicates. This result can be regarded as an example of the fact that coordination numbers and valence indices are not related in a simple manner. The coordination number gives a graphic idea to identify the number of bonds present around an atom while the valence index is an electronic magnitude which indicates the electron vacant sites that an atom can fill in the molecular system. F atoms have valence numbers slightly higher than 1 for both partitionings, which indicates that F atoms also possess an outer shell expansion. Free valences are low for all atoms.
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4. Final remarks and conclusions The valence concepts described in Section 1 are associated to hole distributions and arise from the fundamental expressions in fermion algebra, i.e., the anticommutation creation/annihilation relationships. In this sense, the hole distribution can be considered as the counterpart of the particle distribution and thus it has an equivalent hierarchical value and allows a suitable partitioning. The present model to describe and to determine atomic valences in molecular systems is completely valid within the nonrelativistic quantum physics and so it is suitable for any quality of state function used to describe the system as can be seen from its derivation. Moreover, the set of systems considered to test this formulation is wide enough to include different bonding situations: radicals, excited and hypervalent molecules, the numerical calculations fit well to the expected values for most of them and also show the expected differences between the reported partitioning schemes. It leads to conclude that although valence has been classically defined for closed shell molecules in its ground state, the model proposed in this Letter can be extended to other situations in which classical theory does not provide a definite answer such as excited and hypervalent systems. Extensions in this direction will be reported in the future [30]. At this moment it is worthwhile to mention a striking and attractive new line of research recently developed which is a direct link between quantum chemistry and condensed matter physics [31–34]. These works share the quantities reported in the present Letter and constitute a natural field of application to more complex systems. Research in this direction is currently carried out in our laboratories and will be reported elsewhere [35].
Acknowledgements R.C.B. acknowledges grants in aid from the University of Buenos Aires (project No. X-119) and Consejo Nacional de Investigaciones Cientıficas y Tecnicas, Rep ublica Argentina (PIP No. 02151/01). L.L. and A.T. thank DGI (Spain) and the Universidad del Pais Vasco for their sup-
port with the projects No. BQU 2000-0216 and 00039.310-EB 7730/2000, respectively.
Appendix A. u matrix structure The u matrix can be split into two contributions, one related to the spin density of the electron cloud and the other one to the nonuniform occupancy of the molecular orbitals. In matrix notation u reads u ¼ 21 D 1 D 2 :
ðA:1Þ
The formulation of the one-particle reduced density matrix as the sum of two spin blocks which correspond to spin up (a) and spin down (b), respectively, 1 D ¼ 1 Da þ 1 Db , leads to 2
u ¼ 21 Da þ 21 Db ð1 Da þ 1 Db Þ :
ðA:2Þ
The nonuniformity of each spin block of the oneparticle reduced density matrix, i.e., its lack of 2 idempotency, is formulated as 1 Dr ¼ 1 Dr þ 12Dr , r where r ¼ a or b and D matrix stands for the deviation of idempotency of the Hartree–Fock matrices. The substitution of this formula in Eq. (A.2) yields u ¼ ð1 Dspin Þ2 D; 1
spin
1
ðA:3Þ a
1
b
where D ¼ D D is the spin reduced density matrix and D ¼ Da þ Db collects the total deviation from idempotency as contributions of the spin up (a) and the spin down (b) clouds, respectively.
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