Topography of molecular scalar fields. II. An

Topography of molecular scalar fields. II. An appraisal of the hierarchy principle for electron momentum densities P. Balanarayan and Shridhar R. Gadre Citation: J. Chem. Phys. 122, 164108 (2005); doi: 10.1063/1.1883168 View online: http://dx.doi.org/10.1063/1.1883168 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v122/i16 Published by the AIP Publishing LLC.

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THE JOURNAL OF CHEMICAL PHYSICS 122, 164108 共2005兲

Topography of molecular scalar fields. II. An appraisal of the hierarchy principle for electron momentum densities P. Balanarayan and Shridhar R. Gadrea兲 Department of Chemistry, University of Pune, Pune 411 007, India

共Received 16 August 2004; accepted 7 February 2005; published online 27 April 2005兲 The previously observed hierarchy principle for nondegenerate critical points 共CPs兲 of the electron momentum density 共EMD兲 of molecules 关Kulkarni, Gadre, and Pathak, Phys. Rev. A. 45, 4399 共1992兲兴 is verified at a reliable level of theory. Application of Morse inequalities and the Poincaré– Hopf relation to EMD leads to some rigorous results viz 共i兲 for total number of CPs, NCP = 3 , 7 , 11, 15, . . . there must be either a 共3 , + 3兲 or a 共3 , −1兲 CP at the center of symmetry, 共ii兲 for NCP = 1 , 5 , 9 , 13, . . . there must be either a 共3 , −3兲 or a 共3 , + 1兲 CP at the center of symmetry. A single directional maximum on every ray, starting from p = 0 has been observed for all the test molecules and is suggested as a working topographical principle in p space. This working principle is shown to satisfy the sufficiency condition for the hierarchy principle. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1883168兴 I. INTRODUCTION

Three-dimensional 共3D兲 atomic and molecular scalar fields have become a subject of considerable discussion in physical sciences. These scalar fields enable the extraction of information from the 6N dimensional wave function in a form that is readily interpretable with reference to physicochemical properties of a molecule. Furthermore, some scalar fields such as the electron density 共MED兲 and molecular electrostatic potential 共MESP兲 are accessible through X-ray diffraction experiments. Topography mapping,1,2 involving the identification and characterization of critical points 共CPs兲, enables the physicist and chemist to condense the information succinctly even for large systems. The first part in this series of articles on topography of scalar fields has presented a new algorithm to map the topography of molecular scalar fields3 and applied it to the two scalar fields, viz. MESP and MED. The current work is devoted to the analysis of a much less studied, but nevertheless interesting, scalar field of electron momentum density 共EMD兲. EMD, ␥共p兲, a complementary entity to the coordinate space density ␳共r兲, is directly observable by gamma ray scattering, positron annihilation and 共e , 2e兲 experiments.4 It is known that these two probability densities in conjugate spaces do not bear any direct relationship with each other. EMD is defined via the wave function in momentum space ⌽共p1 , p2 , p3 , . . . , pN兲 as 共the summation over spins being implicit in the equation兲:

␥共p兲 = N

冕

⌽*共p,p2,p3, . . . ,pN兲

⫻⌽共p,p2,p3, . . . ,pN兲d3 p2d3 p3 . . . d3 pN .

共1兲

The momentum space wave function ⌽ is defined as the 3N a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

0021-9606/2005/122共16兲/164108/9/$22.50

dimensional Fourier–Dirac transformation of its position space counterpart ⌿共r1 , r2 , r3 , . . . , rN兲 as ⌽共p1,p2,p3, . . . ,pN兲 = 共2␲兲−3N/2

冉

冕

⫻exp − i

⌿共r1,r2,r3, . . . ,rN兲

N

pj · rj 兺 j=1

冊

d 3r 1d 3r 2 . . . d 3r N .

共2兲

EMDs have been a topic of interest in chemistry since the pioneering work by Coulson advocating the bonddirectionality principle and the analysis of EMD for H2 as well as hydrocarbons.5 Several studies of molecular EMDs, catalyzed by the developments in experimental methods, have been reported during the last six decades. A recent review article by Thakkar6 summarizes the work done on these one electron momentum densities. The interpretation of EMD is rendered difficult due to the absence of nuclei-centric structure seen in position space 共for the more familiar scalar fields of MED and MESP兲. EMD is generally termed as a single-centered distribution and is inversion symmetric around p = 0. Löwdin has shown that for a real N-particle Hamiltonian in position space which has a real eigenfunction, the corresponding momentum space eigenfunction conforms to the property ⌽*共p1 , p2 , p3 , . . . , pN兲 = ⌽共−p1 , −p2 , −p3 , . . . , −pN兲 yielding the inversion symmetry of EMD,7,8 ␥共p兲 = ␥共−p兲. Smith and co-workers9 utilized mathematically and showed graphically the extra symmetry in the momentum density for linear molecules. This feature led to a comparison of the contour plots of molecular EMDs and their position space counterparts by Rawlings and Davidson observing a symmetry enhancement in momentum space densities.10 Gadre and co-workers11,12 put forth a generalization to the symmetry enhancement in momentum space by introducing point groups in momentum space obtained by taking a direct product of the correspond-

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P. Balanarayan and S. R. Gadre

TABLE I. A depiction of the hierarchy principle of CPs of molecular EMD. See Ref. 12 for details. CP at p = 0

CP found elsewhere

共3 , + 3兲共3 , + 1兲共3 , −1兲共3 , −3兲

共3 , + 1兲, 共3 , −1兲, 共3 , −3兲共3 , −1兲, 共3 , −3兲共3 , −3兲 No other CP found

number of types of critical points as given by the Morse inequalities. Further, a general observation that only a single maximum occurs on a ray in momentum space, emanating from p = 0, is presented and its topographical repercussions are discussed. II. METHODOLOGY A. Topography mapping of EMD

ing point symmetry group in position space and the inversion operator i. Yet another typical feature of EMD is the ⬃p−8 fall-off13 of ␥共p兲 for large 兩p兩 values in contrast to the asymptotic exponential decay of ␳共r兲. More recently, Kulkarni, Gadre, and Pathak14 extensively mapped and examined the topography of EMD of a variety of molecules. The inversion symmetric nature of the scalar field requires the point p = 0 to be a critical point. This nondegenerate CP at p = 0 is unique in its nature as determined by the value of EMD and its signature. A nondegenerate CP 共none of the eigenvalues of the Hessian at CP is zero兲 is denoted as 共R , S兲, where R 共rank兲 is the number of nonzero eigenvalues of the Hessian matrix at the CP and S 共signature兲 is the algebraic sum of the signs of the eigenvalues. This implies that 共3 , −3兲 maxima, 共3 , −1兲 and 共3 , + 1兲 saddles and 共3 , + 3兲 minima are the only types of nondegenerate CPs for a 3D scalar field such as EMD. The CPs occurring elsewhere follow a hierarchy14 as shown in Table I with the signature decreasing with respect to the CP at p = 0. All types of CPs as tabulated are invariably found at various values of momenta. The CP at p = 0 is thus a harbinger of the complete EMD topography. When a 共3 , + 3兲 is at p = 0, all the other types of CPs 关共3 , + 1兲, 共3 , −1兲, and 共3 , −3兲兴 are found elsewhere. For a 共3 , + 1兲 at p = 0, 共3 , −1兲 and 共3 , −3兲 CPs and for a 共3 , −1兲 CP at p = 0 only 共3 , −3兲 CPs occur at other p. Kulkarni et al.12 have related the eigenvalues of Hessian at p = 0 to bond polarities in molecules. The above-mentioned principle was put forth by observing the topography at HF/4-31G level, employing the wave functions tabulated by Snyder and Basch.15 Later works by Kulkarni et al.16 and Thakkar et al.17 have examined how the nature and signature of the CP varies with the basis set and amount of electron correlation in the calculation. A review by Thakkar18 elaborates on the properties and anisotropies of EMD at the zero-momentum CP. The topography of EMD was further followed up by a study of catastrophes involved in the analysis of reactions in momentum space by Kulkarni and Gadre.19 The hierarchy principle has thus remained a mystery and why does such a peculiar ordering of CPs exist in momentum space is a question worth examining. The current work is an attempt to further validate and seek an explanation of the above-stated hierarchy principle. The topography of a wide variety of systems both at a higher basis as well as level of theory is mapped utilizing a suitably modified version of the algorithm put forth by the authors in the first paper of this series.3 An appraisal of the hierarchy principle is then attempted through various properties of the inversion symmetric EMD, the asymptotic decay of EMD, and bounds on the

Molecules of diverse chemical nature viz. water 共H2O兲, ethylene 共C2H4兲, cubane 共C8H8兲, tetrahedrane 共C4H4兲, benzene 共C6H6兲, naphthalene 共C10H8兲, anthracene, triphenylene, nitrate anion 共NO−3 兲, borazine 共B3N3H6兲, BN- naphthalene 共B5N5H8兲, a porphyrin derivative and a weakly interacting cluster of ethanediol…6 water, have been chosen as test systems. Their geometries are optimized at HF/ 6-31G** level and the CPs mapped by invoking a modified version of the algorithm put forth in Ref. 3. The EMD has been evaluated on a single spherical surface centered at p = 0 and the surface maxima, minima, and saddles were found. EMD evaluation has been done utilizing the property package INDPROP20 from the wave function of the system obtained by the 21 22 GAUSSIAN94 /GAMESS suites of programs. The function is evaluated on rays through the surface maxima, minima and saddles, starting from p = 0 to 兩p兩 = 4 a.u. The rays are bisected to give directional stationary points, which are taken as guess points for a Newton–Raphson optimization routine involving the analytical Hessian at the point. An alternative optimizer of STEPIT23 involving a cyclic relaxation technique is also used. The search is carried out until a convergence threshold 1.0⫻ 10−10 is achieved.3 The CPs, their nature, and the corresponding EMD values are tabulated in Table II. The figures presented here are generated using the in-house developed visualization package24 UNIVIS-2000. B. Appraisal of the hierarchy of EMD CPs

The appraisal of the hierarchy of CPs involves a stepwise application of the following properties of the scalar field. (i) Inversion symmetry around p = 0: The inversion symmetry of EMD, as discussed in Sec. I is binding on the number and types of the CPs found in EMD. Since the point at p = 0 has to be a CP because of inversion symmetry, the total number of CPs, viz. NCP has to be odd in number, where n−3 + n−1 + n+1 + n+3 = NCP .

共3兲

Here, the notation 共n−3 , n−1 , n+1 , n+3兲 lists successively the number of maxima, 共3 , −1兲 saddles, 共3 , + 1兲 saddles and minima. Furthermore, the CPs have to reflect the p-space molecular symmetry. The inversion symmetry property results into a constraint that not more than one type of CP can appear odd number of times. (ii) Poincaré–Hopf relation: The relation for a scalar field with an asymptotic fall-off at the boundaries is given by n−3 − n−1 + n+1 − n+3 = 1.

共4兲

The above-mentioned relationship in conjunction with Eq. 共3兲, leads to the following constraints:


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Hierarchy principle in momentum space

TABLE II. Topography of various test molecules with the number and EMD values in a.u. at the CPs in parentheses at HF/ 6-31G** level of theory. Here n+3 is the number of minima n+1 the number of 共3 , + 1兲 saddles, n−1 the number of 共3 , −1兲 saddles, and n−3 the number of maxima. See the text and figures for details. System Water Ethanediol-6 water Benzene Naphthalene Ethylene Nitrate anion Cubane Tetrahedrane Borazine BN-naphthalene Triphenylene Porphyrin derivative Anthracene

n−3 + n+1 = 共NCP + 1兲/2,

n+3

n+1

n−1

n−3

1共0.548兲 1共6.079兲 1共4.133兲 0 0 1共1.438兲 0 0 0 0 0 0 0

2共0.595兲 4共6.162兲 8共4.140,4.282兲 0 1共2.113兲 8共1.485,1.488,1.487兲 0 0 0 0 0 1共25.924兲 0

2共0.635兲 4共6.238,6.177兲 18共4.141,4.287兲 1共6.276兲 2共2.114兲 10共1.487,1.490,1.543兲 0 0 1共4.510兲 1共6.671兲 1共10.165兲 2共25.925兲 1共8.414兲

2共0.659兲 2共6.345兲 12共4.287兲 2共6.556兲 2共2.165兲 4共1.546兲 1共5.590兲 1共3.115兲 2共4.511兲 2共6.702兲 2共10.811兲 2共27.320兲 2共8.793兲

n+3 + n−1 = 共NCP − 1兲/2.

共5兲

25

(iii) Weak Morse inequalities. The following inequalities apply to the individual numbers of different types of nondegenerate CPs and basically follow from connectivities of CPs of a three-dimensional scalar field asymptotically decreasing/decreasing at the boundaries. The boundaries are chosen such that CPs on it are not counted in the inequalities and the bounding surface is orientable, i.e., the normal to the surface element has a continuous direction everywhere. The weak Morse inequalities are expressed by the following relations: n−3 艌 1,

n−3 − n−1 艋 1,

n−3 − n−1 + n+1 艌 1, n+3 − n+1 艋 0,

n+3 艌 0,

共6兲

n+3 − n+1 + n−1 艌 0.

(iv) Nature of criticalities and subspaces bound by a zero flux surface: Every maximum in a scalar field is bound by a minimal surface. The CPs that have a minimal characteristic, as indicated by a positive eigenvalue of the Hessian matrix, form a part of the minimal surface. If this surface is taken as a boundary then the CPs on it are not accounted for in the Morse inequalities. Thus if there exists only one 共3 , −3兲 maximum and at least one of the other types of CPs then we would have more than one subspace. Now there would exist subspaces without maxima and this violates the definition of a subspace for a function which dies off asymptotically. The schematic in Fig. 1 shows the kind of arrangement of CPs

that is not valid as stated earlier. Thus a single maximum with all other CPs is a case which is not possible by the first Morse inequality applied to subspaces which decrease at the boundaries. Nevertheless, an arrangement of the above with more than one strict maximum and other types of CPs is still valid. The above four rigorous conditions are applied to an algebraic permutation of numbers of different types of CPs with the total number of CPs, NCP = 1 , 3 , 5 , 7 , . . . . The resultant possibilities are displayed in Table III. Apart from the above-mentioned conditions, a working principle found out from a detailed observation of the EMD has also been employed in the forthcoming analysis. (v) Single maximum on a ray starting from p = 0: The following characteristics are observed for molecular EMDs. If there is a directional maximum at p = 0 then there does not exist a directional criticality anywhere else on the ray and the function goes to zero monotonically. If there is a directional minimum at p = 0, then the function attains one and only one directional maximum at 兩p兩 ⬎ 0 and thereafter dies to zero monotonically. This condition may be taken as a working principle that has been observed from the analysis of all the molecular EMDs. However, this condition remains unproven analytically for EMD in a general case. A further justification for this is provided by an examination of the momentum densities5 of H2 molecule within simple VB and MO frameworks. The EMD of H2 molecule within VB and MO theory are given, respectively, as FIG. 1. See the text for details. 共a兲 Schematic showing the partitions of the asymptotically decaying nonnegative scalar field when there is only one 共3 , −3兲 and all other CPs. This arrangement is not valid since in the partitions where there are no maxima the first Morse inequality is not obeyed in the subspace. 共b兲 A subspace partitioning-wise valid arrangement of CPs.


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TABLE III. Various possibilities for number of CPs, NCP = 1 , 3 , 5 , 7 , 11, . . ., generated by applying conditions 共i兲–共iv兲. See the text 共Secs. II and III兲 for details. The notation follows the order 共n−3 , n−1 , n+1 , n+3兲, which are the number of maxima, number of 共3 , −1兲 CPs, number of 共3 , + 1兲 CPs, and number of minima, respectively.

NCP

Number of arrangements

1 3 5 7 9

1 1 2 4 6

11

9

13

12

15

16

共n−3 , n−1 , n+1 , n+3兲共1, 共2, 共3, 共4, 共5, 共2, 共6, 共2, 共2, 共7, 共4, 共3, 共8, 共4, 共4, 共2,

0, 1, 2, 3, 4, 2, 5, 1, 5, 6, 4, 6, 7, 3, 7, 4,

0, 0, 0, 0, 0, 3, 0, 4, 4, 0, 3, 4, 0, 4, 4, 6,

for molecules, even those which contain these Type 3 atoms such as ZnO and ZnCl2. The nonspherically averaged ground states of atoms O, C, N, V, and Ti are observed to obey the single ray maximum principle in their 3D momentum densities. III. RESULTS AND DISCUSSION

0兲 0兲 0兲, 共2, 2, 1, 0兲 0兲, 共2, 2, 2, 1兲, 共2, 3, 2, 0兲, 共2, 1, 2, 2兲 0 兲, 共4, 4, 1, 0兲, 共3, 2, 2, 2兲, 共3, 4, 2, 0兲 2兲, 共2, 4, 3, 0兲 0兲, 共4, 3, 2, 2兲, 共4, 4, 2, 1兲, 共4, 5, 2, 0兲 4兲, 共2, 2, 4, 3兲, 共2, 3, 4, 2兲, 共2, 4, 4, 1兲 0兲 0兲, 共6, 6, 1, 0兲, 共5, 4, 2, 2兲, 共5, 6, 2, 0兲 2兲, 共4, 6, 3, 0兲, 共3, 2, 4, 4兲, 共3, 4, 4, 2兲 0兲, 共2, 2, 5, 4兲, 共2, 4, 5, 2兲, 共2, 6, 5, 0兲 0兲, 共6, 5, 2, 2兲, 共6, 6, 2, 1兲, 共6, 7, 2, 0兲 4兲, 共4, 4, 4, 3兲, 共4, 5, 4, 2兲, 共4, 6, 4, 1兲 0兲, 共2, 1, 6, 6兲, 共2, 2, 6, 5兲, 共2, 3, 6, 4兲 3兲, 共2, 5, 6, 2兲, 共2, 6, 6, 1兲, 共2, 7, 6, 0兲

␥共p兲 =

1 + S cos关p · 共ra − rb兲兴兩A共p兲兩2 , 1 + S2

共7a兲

␥共p兲 =

1 + cos关p · 共ra − rb兲兴兩A共p兲兩2 , 1+S

共7b兲

where 共ra − rb兲 is the bond vector, A共p兲 is the Fourier transform of the 1s atomic Slater function 共␣3/2 / ␲1/2兲exp共−␣r兲, and S is the overlap between the two AOs. For all the rays, a unique solution is seen at p = 0, which is always maximal. It is to be noted that this feature is present only for a range of optimal values of S and ␣ 共here we have used S = 0.624, ␣ = 1.18 at RAB = 0.72 Å兲. Any arbitrary combination of S and ␣ may not have a unique maximum. This suffices to show that just the topographical features 共Morse inequalities, Poincaré– Hopf and all the rest of the conditions discussed earlier兲 do not make the analysis complete. This ray condition is observed for all the Hartree–Fock 共HF兲 level EMDs of the calculated test systems at all the basis sets and even with methods that include correlation such as Becke three parameter exchange and Lee–Yang–Parr 共B3LYP兲 correlation density functional method or Møller–Plesset second-order 共MP2兲 perturbation theory. It is to be noted that this working principle is not valid for spherically averaged momentum densities, which do possess multiple maxima on the ray.26 Koga et al.26 have thus classified all the atoms in their ground state based on modalities of the spherically averaged momentum densities into three types of which the third type consists of two maxima on a ray. For the atoms Zn, Ag, Cd, Au, and Hg 共called Type 3 atoms兲 that have spherically symmetric ground states, the second ray maxima appear as degenerate maxima and not strict isolated 共3 , −3兲 CP. Hence, in these cases the EMD is not a Morse function.25 It may be noted that hierarchy principle 共see Table I兲 is still valid since it deals with only isolated nondegenerate CPs as mentioned before. The single ray maximum principle is seen to be valid

A. Topography of EMD and verification of the hierarchy principle

The results for all the test cases reported in Table II are in conformity with the hierarchy principle in p space, discussed before. Although the nature of the topography as given by the number of CPs and their arrangements is seen to differ from the previous observations12,14,16 for some systems such as benzene, the hierarchy is still maintained irrespective of the basis set used and the level of theory. The test cases of water, naphthalene, ethylene, and cubane are a few whose EMDs possess minimal set of CPs 共the least number of each type of CP兲 that satisfies the hierarchy principle. EMD of water 共see Figs. 2–4兲 displays a minimum at p = 0 followed by two 共3 , + 1兲’s, two 共3 , −1兲’s, and two 共3 , −3兲 CPs. The values of EMD at the CPs are found to be between 0.5 and 0.6 a.u., which are rather low compared to the EMD values for the other test cases. All the CPs lie within a sphere of radius 兩p兩 = 1.5 a.u. Yet another feature worth noting is that the EMD CPs for water are more in number as compared to the corresponding number for MED. The EMD of naphthalene 共XY is the molecular plane for position space, see Fig. 3兲 shows a 共3 , −1兲 CP at p = 0 followed by two maxima along pz axis, one of which is seen on the plane pz = 0.34 a.u. in Fig. 3共b兲. This may perhaps be looked upon as a manifestation of aforementioned bond directionality principle for a polyatomic system. All the planar test cases 共except for water兲 show maxima in their EMDs, which lie on p-space directions that are perpendicular to the coordinate space plane containing the bonding regions. This specific directional preference is not observed for systems which are cage-like in position space such as cubane and tetrahedrane. A rich topography, in comparison to all the other test cases, is observed for benzene 关see Fig. 5共b兲兴 with 39 CPs. The arrangement involves a 共3 , + 3兲 at p = 0 with all the other kinds of CPs arranged in accordance with the p space symmetry of benzene. The EMD CPs of the nitrate anion 关Fig. 5共b兲兴 belong to the same hierarchy classification as that of benzene and again a large number of CPs 共23兲 is observed. The weakly bonded cluster ethanediol with 6 water molecules 关see Fig. 5共c兲兴 with 11 CPs also has a 共3 , + 3兲 CP at p = 0. Borazine 共termed “inorganic benzene”兲 is a system isoelectronic with benzene. Even the geometries of benzene and borazine are similar and one finds a comparison of their aromatic stabilities in the literature. However, there is a vast difference in the EMD topography of these systems 共Fig. 4兲. Borazine is endowed with just 3 CPs with a 共3 , −1兲 CP at p = 0 and two maxima elsewhere. The EMD values at the CPs are very similar to that for benzene. BN-naphthalene, triphenylene, and anthracene 共refer to Table II兲 exhibit topog-


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FIG. 2. 共a兲 EMD topography of water at HF/ 6-31G** level of theory. 共b兲 EMD of the water molecule textured on planes px = 0, py = 0, pz = 0.

raphy similar to that of borazine except for the EMD values at the CPs changing due to the number of electrons in each of the system. The separation of the CPs over the range of momenta is seen to decrease with increasing size. The effect of correlation on the topography has been examined by mapping the topographies of benzene and borazine for STO-3G, 6-31G, 6-31G**, and 6-311+ + G** basis sets at HF, B3LYP, and MP2 levels of theory employing 21 GAUSSIAN 94 package. The topography of benzene at STO-3G basis reveals only one 共3 , −3兲 CP at p = 0. The topography at all the other levels of theory and basis sets is seen to consist of 39 CPs as mentioned in the preceding paragraph. The EMD values at the CPs do change but the change is not significant. Borazine does not show any qualitative change in the topography at all the above-mentioned basis sets and levels of theory. In addition, the hierarchy principle is maintained in all the cases.

The EMD of all these systems is seen to obey condition 共v兲 i.e., single directional maximum on a ray, which has been taken as a working principle for an appraisal of the hierarchy principle. The following discussion attempts to offer some physical and mathematical reasoning for the observed hierarchy of CPs in EMD. B. Appraisal of hierarchy principle

Table III lists the permutations of the numbers of different types of CPs for NCP after the application of conditions 共i兲–共iv兲 described in Sec. II. The following results are arrived at by an examination of the tabulated arrangements of CPs: 共a兲共b兲

For NCP = 3 , 7 , 11, 15, . . ., there must be either a 共3 , + 3兲 or a 共3 , −1兲 CP at p = 0. For NCP = 1 , 5 , 9 , 13, . . ., there must be either a 共3 , −3兲 or a 共3 , + 1兲 CP at p = 0.

FIG. 3. 共a兲 EMD topography of naphthalene at HF/631G** level of theory. 共b兲 EMD of naphthalene textured on planes px = 0, py = 0, pz = 0.34 a.u.


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FIG. 4. 共a兲 EMD topography of ethylene at HF/631G** level of theory. 共b兲 EMD of ethylene textured on planes px = 0, py = 0, pz = 0.

共c兲

There cannot be a unique maximum at p = 0 with other types of CPs elsewhere, a result directly stems from condition 共iv兲.

The above-mentioned statements are applicable to not only the topography of EMD but for any inversion symmetric asymptotically decaying scalar field. The listed configurations of CPs in Table III could as well exist for an inversion symmetric position space density. For NCP = 1, there is only one arrangement possible, viz. a 共3 , −3兲 maximum. This result is clearly evident from the first Morse inequality in Eq. 共6兲. This topography is observed for cubane and tetrahedrane. In the discussion hereafter, the arrangement of CPs is denoted as a quartet 共n−3 , n−1 , n+1 , n+3兲. For NCP = 3, the only possible arrangement using conditions 共i兲–共iv兲, is 共2, 1, 0, 0兲. As seen from Table III for NCP = 5 , 7 , 11, . . . there do exist arrangements of CPs that do not follow hierarchy principle. The case NCP = 7 is discussed here in detail. After the application of conditions 共i兲–共iv兲, there are still four arrangements possible which are 共4, 3, 0, 0兲, 共2, 2, 2, 1兲, 共2, 3, 2, 0兲, and 共2, 1, 2, 2兲. Out of these four arrangements only one, viz. 共2, 2, 2, 1兲

follows hierarchy principle. These arrangements are such that there is a 共3 , −1兲 CP at p = 0 and one arrangement with a 共3 , + 3兲 at p = 0. All the possible geometric arrangements are sketched schematically in Fig. 5. employing the following conventions and rules. 共1兲

The arrows connecting the CPs are representative of the directional derivative along any ray given by p · ⵱ ␥共p兲 gradient paths along a ray emanating from p = 0. 共2兲 The directions of the schematic paths are according to the conventional definition of a gradient of a function that always points in the direction of increase. 共3兲 The existence of at least one such path connecting “adjacent” CPs is the basis of sketches. 共4兲 The depicted connectivities are those for the CP at p = 0. These geometric arrangements follow from the consideration that every pair of maxima 共neighboring and connected兲 are linked by a 共3 , −1兲, pairs of minima linked by 共3 , + 1兲’s or 共3 , −1兲’s, and the saddles are linked similarly.27 It is understood that the real gradient paths could be

FIG. 5. EMD topographies of 共a兲 nitrate anion, 共b兲 benzene, and 共c兲 Ethanediol…6 water.


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FIG. 6. Different arrangements 共n−3 , n−1 , n+1 , n+3兲 of CPs through schematic directional derivative gradient path connectivities around p = 0 for the case NCP = 7, for arriving at hierarchy principle through conditions 共i兲– 共iv兲 and the working principle of a single ray maximum. Cases 共a兲共4, 3, 0, 0兲共d兲共2, 3, 2, 0兲 and 共e兲 and 共f兲共2, 1, 2, 2兲 violate single directional maximum on a ray. Cases 共b兲 and 共c兲 violate the definition of a 共3 , −1兲 CP. Case 共g兲 shows the valid arrangement that follows hierarchy.

curved and need not be along the orthogonal axes of the coordinate system in momentum space but the directional derivative gradient paths are directed along a ray emanating from p = 0. The analysis of these schematic connectivities is done by invoking the working principle of a single directional ray maximum. Figure 6共a兲 depicts the case of 共4, 3, 0, 0兲. This arrangement is not observed in momentum space topography since it violates the adopted working principle of a single ray maximum. The depicted arrangement is observed for inversion symmetric position space densities 共e.g., linear molecules such as acetylene, systems with non-nuclear maxima, etc.1兲. Figures 6共b兲 and 6共c兲 show schematics for the same 共4, 3, 0, 0兲, but these clearly would not exist for any inversion symmetric scalar field since the connectivities violate the definition of a 共3 , −1兲 CP which is a maximum in two directions and a minimum in one direction. Reversing the arrows for these two structures would still violate general topographical rules since then the CPs connected to it are maxima and, by definition, they cannot be lower in magnitude than the 共3 , −1兲 at p = 0. The rest of the sketches are such that the definition of the CP at p = 0 is not violated. Hence, they are possible arrangements of CPs for a general inversion symmetric scalar field but not momentum density since the single ray maximum is violated in each case. Figure 6共g兲 displays the 共2, 2, 2, 1兲 case with a minimum at p = 0 wherein all the aforementioned rules are obeyed and it is the only arrangement compatible with hierarchy principle. EMD of water is an example of the topography depicted in Fig. 6共g兲. A similar analysis when applied to all the other arrangements for NCP = 1 , 3 , 5 , 7 , 11, . . ., rules out all possibilities which do not follow hierarchy and the deciding factor is the working principle which states that there is only a single

directional maximum either at p = 0 or elsewhere. The above pictorial description may be alternatively brought out by the following analysis. Arriving at a sufficiency condition for the hierarchy principle: The single ray maximum principle combined with conditions 共i兲–共iv兲 give the following results on the connectivities of CPs. Case 1: 共3 , −3兲 CP at p = 0 (maximal in all directions). No other CP can be connected to it due to the ray maximum principle. Therefore NCP = 1. Case 2: 共3 , −1兲 at p = 0 (maximal in all directions except one). 共1兲共2兲

共3兲共4兲

共5兲

Only two equivalent inversion symmetric maxima can be connected to it. A pair of 共3 , −1兲 CPs cannot be connected to it in such a way that the connected CPs are maximal in ray direction because then the definition of a 共3 , −1兲 CP being a minimum in only one direction is violated. 共3 , + 3兲 cannot be connected to it just by the unique ray maximum observation. Hence n+3 = 0. A pair of 共3 , + 1兲 saddles can be connected to it in such a way that the connected CPs are oriented in such a way that they are directional maxima on the ray. However this case can never be seen because of the Morse inequality n+3 − n+1 艌 0 as given in Eq. 共6兲 and the statement made above 共see n+3 = 0兲. Hence n+1 = 0. Therefore we have the result that the total number of CPs when a 共3 , −1兲 is at p = 0, would always be 3 共2 maxima and the CP at p = 0兲.

Case 3: 共3 , + 1兲 at p = 0 (minimal in all directions except one).


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FIG. 7. Schematic cross-sectional depiction of the zeroflux surfaces for EMD for all four cases in the hierarchy principle 共cf. Table I兲. The solid line represents the zero-flux surface. The dashed line depicts the zero-flux surface at infinity 共see the text for details兲.

共1兲 Any number of pairs of maxima and 共3 , −1兲 CPs can be connected to it. 共2兲共3 , + 3兲 cannot be connected to it just by the unique ray maximum observation. Hence n+3 = 0. 共3兲 As mentioned for the case of a 共3 , −1兲 CP at p = 0 the above-presented statement ensures that there are no 共3 , + 1兲 CPs connected to the 共3 , + 1兲 at p = 0. 共4兲 Thus NCP = 1 + n−3 + n−1. Case 4: 共3 , + 3兲 at p = 0 (minimal in all directions). All the types of CPs could be connected to it in such a way that they are all directional ray maxima, yielding NCP = 1 + n−3 + n−1 + n+1. All the four cases above when applied to Table III leave us with only arrangements that are in conformity with the hierarchy principle that was previously stated as given in Table I. Thus the single ray maximum principle provides a sufficiency condition for the hierarchy principle. The hierarchy principle could be restated as unique directional maximum on all the rays emanating from p = 0. Further implications of the ray maximum sufficiency condition and the hierarchy principle: The topological partitioning in r space is readily visualizable because all nuclei appear as maxima. EMD does not contain any “nuclear information,” but the hierarchy principle and the unique ray maximum bring out the nature of topological partitioning in p space. The partitioning of EMD could be achieved through a zero-flux surface defined by the condition ⵜ␥共p兲 · dS = 0, dS = nˆ PdS. This surface would never exist perpendicular to a ray in momentum space since the function never goes minimal along a ray. The surface element dS would always be parallel to a ray p 共from p = 0兲 in momentum space and perpendicular to it only at infinity where the function goes minimal. Maxima which are attractors in topological language1 would be near the function weighted center of the subspace bounded by the above-mentioned zero-flux surface that passes through all the minima 共repellers in topological language兲 and the 共3 , −1兲 and 共3 , + 1兲 saddles. The schematic possibilities of the zero-flux surfaces are depicted in Fig. 7. For the case where there is a 共3 , −3兲 CP at p = 0 the zero-flux surface exists only at infinity and hence there is only a single inversion symmetric subspace which is the whole space itself. A 共3 , −1兲 CP at p = 0 indicates a single planar zero-flux surface which divides the whole space into two equivalent subspaces which are individually not inversion symmetric. A 共3 , + 1兲 CP at p = 0 would have noninversion symmetric subspaces equal to the number of maxima in its vicinity. The same could be said for the case where a minimum exists at p = 0. The schematic diagrams are given for the least number of each type of CP satisfying hierarchy principle and the extension to a greater number of CPs is easily done using the

aforementioned rules. A p-space topological subspace in general contains information about the properties of more than one atom. The subspace partitioning of EMD seems to be of chemical interest and warrants further examination. IV. CONCLUDING REMARKS

To summarize, a detailed scrutiny of the hierarchy principle of CPs in EMD is carried out by mapping the topography for several systems of diverse chemical nature at a good quality basis and level of theory. It is seen that the inclusion of correlation, although altering the nature of the criticality, still does not violate the hierarchy principle. The appraisal presented here is an attempt to explain the hierarchy principle observed in momentum space through topological inequalities and inversion symmetry. These engender some rigorous results, which are valid for any inversion symmetric scalar field. A further step is taken in analyzing the hierarchy principle by invoking a working principle from an observation that a ray for molecular EMD contains just one directional maximum. This working principle is shown to satisfy the sufficiency condition for the hierarchy principle and is unique to molecular momentum space densities in contrast to other 3D molecular scalar fields such as MED or MESP. The repercussions of the single ray maximum are discussed in view of the topological partitioning of momentum densities. The single ray maximum observation needs to be examined further rigorously: its manifestation in the EMDs through the Schrödinger equation in p space is perhaps the solution to the remaining part of the riddle, and this is an ongoing work in our laboratory. The subspace partitioning of momentum space densities and what do the subspaces in EMD imply is another interesting question to be examined. Position space density has had the convenience of a chemically intuitive nuclear-centered distribution. With EMD now picking up as a very “comfortable” index to be handled for molecules,28,29 questions such as “why its topography is the way it is” need to be answered. It is hoped that the current work forms a step in this direction. ACKNOWLEDGMENTS

P.B. acknowledges the Council for Scientific and Industrial Research 共CSIR兲 New Delhi, for the award of a research fellowship. The authors are indebted to Dr. Sudhir A. Kulkarni, Vlife Sciences, Pune and Dr. Rajeev Pathak, Department of Physics, University of Pune for valuable suggestions and advice in the current work. R. F. W. Bader, Atoms in Molecules: A Quantum Theory 共Oxford University Press, Oxford, 1990兲. 2 S. R. Gadre, in Computational Chemistry: Reviews of Current Trends, 1


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edited by J. Lesczynski 共World Scientific, Singapore, 2000兲 Vol. 4, pp. 1–53. 3 P. Balanarayan and S. R. Gadre, J. Chem. Phys. 119, 5037 共2003兲. 4 I. R. Epstein and A. C. Tanner, in Compton Scattering, edited by B. Williams 共McGraw-Hill, New York, 1977兲. 5 C. A. Coulson, Proc. Cambridge Philos. Soc. 37, 55 共1941兲; W. E. Duncanson and C. A. Coulson, ibid. 37, 406 共1941兲; see also W. E. Duncanson, ibid. 39, 180 共1943兲, and references therein. 6 A. J. Thakkar, Adv. Chem. Phys. 128, 303 共2004兲. 7 P. Kaijser and V. H. Smith, Jr., in Quantum Science: Methods and Structure, edited by J. L. Calais, O. Goscinski, J. Linderberg, and Y. Ohrn 共Plenum, New York, 1976兲. 8 P.-O. Löwdin, in Advances in Quantum Chemistry edited by P.-O. Löwdin 共Academic, New York, 1967兲, Vol. 3, p. 323. 9 A. J. Thakkar, A. M. Simas, and V. H. Smith Jr., J. Chem. Phys. 81, 2953 共1984兲. 10 D. C. Rawlings and E. R. Davidson, J. Phys. Chem. 89, 969 共1985兲. 11 S. R. Gadre, A. C. Limaye, and S. A. Kulkarni, J. Chem. Phys. 94, 8040 共1991兲. 12 S. A. Kulkarni and S. R. Gadre, Z. Naturforsch., A: Phys. Sci. 48, 145 共1993兲. 13 R. Benesch and V. H. Smith, Jr., in Wave Mechanics—The First Fifty Years, edited by W. C. Price, S. S. Chisick, and T. Ravensdale 共Butterworths, London, 1973兲, pp. 357–377. 14 S. A. Kulkarni, S. R. Gadre, and R. K. Pathak, Phys. Rev. A 45, 4399 共1992兲. 15 L. C. Snyder and H. Basch, in Molecular Wave functions and Properties 共Wiley, New York, 1972兲. 16 S. A. Kulkarni and S. R. Gadre, Chem. Phys. Lett. 274, 255 共1997兲. 17 A. J. Thakkar and B. S. Sharma, J. Mol. Struct.: THEOCHEM 527,

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