Topology of electron–electron interactions in atoms and molecules. III. Morphology of
electron intracule density in two 1 g + states of the hydrogen molecule
Jerzy Cioslowski, Guanghua Liu, Jacek Rychlewski, Wojciech Cencek, and Jacek Komasa
Citation: The Journal of Chemical Physics 111, 3401 (1999); doi: 10.1063/1.479624
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 8
22 AUGUST 1999
Topology of electron–electron interactions in atoms and molecules.
III. Morphology of electron intracule density in two 1 ⌺ ⴙ
g states
of the hydrogen molecule
Jerzy Cioslowskia) and Guanghua Liu
Department of Chemistry and Supercomputer Computations Research Institute, Florida State University,
Tallahassee, Florida 32306-3006
Jacek Rychlewski
Quantum Chemistry Group, Department of Chemistry, A. Mickiewicz University, ul. Grunwaldzka 6,
60-780 Poznan, Poland, and Institute of Bioorganic Chemistry, Polish Academy of Sciences, PCSS,
ul. Noskowskiego 12-14, Poznan, Poland
Wojciech Cencek and Jacek Komasa
Quantum Chemistry Group, Department of Chemistry, A. Mickiewicz University, ul. Grunwaldzka 6,
60-780 Poznan, Poland
共Received 4 February 1999; accepted 26 May 1999兲
The differences in electronic structures of two 1 ⌺ ⫹
g states of the hydrogen molecule are vividly
reflected in their intracule densities I(r). The ground-state wave function of H2 is associated with
two distinct topologies of I(r) 共one of which pertains to the united atom limit兲, whereas no fewer
than 11 unequivalent sets of critical entities are found for I(r) of the EF state that involves multiple
electronic configurations. These sets and the catastrophes that interrelate them, which arise from
conflicts between topological features of I(r) pertinent to different configurations, are characterized
in detail. The usefulness of topological analysis of I(r) in the detection and characterization of
various types of electron correlation is demonstrated. © 1999 American Institute of Physics.
关S0021-9606共99兲30132-X兴
I. INTRODUCTION
duced concept stems from the fact that it enables rigorous
analysis of electron correlation effects.3 The cage ⍀ 关 I 兴 constitutes the domain in the space of interelectron distance vectors r within which the correlation effects exhibited in I(r)
are substantial. Integration of observables related to I(r)
over ⍀ 关 I 兴 affords quantitative measures of electron correlation, such as the number of strongly correlated electron pairs
N corr关 I 兴 ,
Topological analysis of the electron intracule density
I(r) 1,2 provides detailed insights into electron–electron interactions in atoms and molecules.3 Topology of I(r) is inherently far more complex than that of the one-electron density
␳共r兲.3–5 First of all, the gradient ⵜI(r) vanishes in systems
with cylindrical and spherical symmetries not only at critical
points but also at critical circles and critical spheres—a feature that, according to the empirical evidence accumulated
thus far, is never exhibited by the ground-state ␳共r兲.6 Second,
attractors in I(r) are usually connected through multiple interaction lines, which are exceedingly rare in one-electron
densities.7
Among the critical entities pertinent to I(r), the 共3,3兲
critical point 共the cage point兲 located at r⫽0 is certainly the
most important. This point arises from the electron–electron
coalescence cusp,8,9 reproduction of which requires either the
inclusion of terms linear in r 12 共⫽兩r兩兲 in the electronic wave
function or the use of the generalized Hiller–Sucher–
Feinberg formalism.9,10 However, even less sophisticated approaches such as MP2 are capable of producing electron intracule densities with minima at the origin.11
The critical entities that surround the cage point are connected by a web of gradient paths that delineate the walls of
the correlation cage.3 The significance of this recently intro-
N corr关 I 兴 ⫽
⍀关I兴
共1兲
I 共 r兲 dr,
and their contribution to the electron–electron repulsion energy W corr关 I 兴 ,
W corr关 I 兴 ⫽
冕
⍀关I兴
I 共 r兲 r ⫺1 dr.
共2兲
The cage volume V corr关 I 兴 characterizes the spatial extent of
electron correlation and the unitless ratio
␬ 关 I 兴 ⫽I 共 0兲 V corr关 I 兴 /N corr关 I 兴 ,
0⬍ ␬ 关 I 兴 ⬍1,
共3兲
measures the degree of variation in the electron intracule
density within ⍀ 关 I 兴 , i.e., the relative reduction in its magnitude as r˜0. Together with the shape of ⍀ 关 I 兴 , these quantities yield the picture of electron correlation that is much
more detailed than that afforded by simple statistical
coefficients.12
It is important to emphasize that the shape of the correlation cage and its spatial extent are entirely determined by
the topological properties of I(r). Thus, unlike that of the
a兲
Author to whom the correspondence should be addressed. E-mail address:
jerzy@kyoko.chem.fsu.edu, homepage: http://www.scri.fsu.edu/⬃jerzy.
0021-9606/99/111(8)/3401/9/$15.00
冕
3401
© 1999 American Institute of Physics
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J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
FIG. 2. The symmetry-unique set of critical entities and the connecting
gradient paths in the electron intracule density of the ground state of H2
共oriented along the z axis兲 at R⫽2.0 关 a.u.兴 .
FIG. 1. The positions of the attractors and the 共2,0兲 critical circle in the
electron intracule density of the ground state of H2 vs R.
Coulomb hole,2,13–15 the definition of ⍀ 关 I 兴 does not rely on
an ‘‘uncorrelated’’ reference that may be ill-defined for systems with significant nondynamical electron correlation
共such as molecules at the dissociation limit兲.
There are very few many-electron systems for which the
accuracy of modern quantum-chemical calculations exceeds
that of experimental measurements. One of these systems is
the hydrogen molecule. Despite the availability of almost
exact electronic wave functions, only rather inaccurate I(r)
of the ground state of H2 have been analyzed thus far,15
while those of the excited states have not been investigated
at all. Prompted by this scarcity of data, we report here a
detailed topological analysis of the electron intracule densities computed from explicitly correlated wave functions of
the ground and the double-minimum EF excited states of H2 .
The present study of these two 1 ⌺ ⫹
g states is intended to
serve as a benchmark for future work involving larger systems.
In Eqs. 共4兲 and 共5兲, the subscripts 1 and 2 denote electrons,
whereas the subscripts A and B refer to the nuclei. The parameters 兵 c k 其 , 兵 ␣ k 其 , 兵 ␤ k 其 , 兵 ␨ k 其 , 兵 ␩ k 其 , and 兵 ␥ k 其 were determined variationally by minimizing the first 共for the ground
state兲 or the second 共for the EF state兲 eigenvalue of the electronic Hamiltonian with the conjugate directions method of
Powell.17,18 The expansion lengths K of 1200 and 600 were
used for the ground and EF states, respectively, resulting in
energies with the extraordinary accuracy of 10⫺10 关 a.u.兴 in
the former case.19 More details on these wave functions and
the algorithms employed in their calculation are available
elsewhere.18
The electron intracule densities derived from the aforedescribed wave functions are expected to be accurate to
within 10⫺5 关 a.u.兴 . Although devoid of electron–electron
II. CALCULATIONS
The spatial components of the electronic wave functions
employed in this work are given by linear combinations of
properly symmetrized Gaussian-type geminals 兵 ␺ k 其 , 16
K
⌿ 共 r1 ,r2 兲 ⫽ 共 1⫹ P̂ 12兲共 1⫹î e 兲
兺
k⫽1
c k ␺ k 共 r1 ,r2 兲 ,
共4兲
where P̂ 12 and î e are the electron exchange and inversion
operators, respectively, and
2
2
2
2
␺ k 共 r1 ,r2 兲 ⫽exp共 ⫺ ␣ k r 1A
⫺ ␤ k r 1B
⫺ ␨ k r 2A
⫺ ␩ k r 2B
FIG. 3. The electron intracule density of the ground state of H2 at 共a兲 the
attractor, 共b兲 the 共2,0兲 circle, and 共c兲 the cage point vs. R.
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2
⫺ ␥ k r 12
兲.
共5兲
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J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
cusps, they exhibit minima at the vanishing interelectron
distance.20 The other critical entities of these densities were
located with a standard algorithm.21 The numerical methods
employed in the tracing of gradient paths and the determination of the cage properties have been described previously.3
III. MORPHOLOGY OF I „r… IN THE 1 1 ⌺ ⴙ
g GROUND
STATE OF H2
The I 1 ⌺ ⫹
g ground state of the hydrogen molecule has
the united atom limit of 1 S(1s 2 ). At that limit, the respective
I(r) possesses a single 共1,⫺1兲 critical sphere with a radius of
0.1937 关a.u.兴 共Fig. 1兲. Surrounding the cage point, this sphere
demarcates the boundary of the correlation cage. Separation
of the hydrogen nuclei triggers a 共⫹兲-cusp catastrophe6,22 in
I(r). The critical sphere is replaced by a set of two attractors
located on the molecular axis of symmetry and one 共2,0兲
critical circle perpendicular to it3 共Fig. 2兲. The plane encompassing this circle, which contains the cage point, divides the
Cartesian space into the basins of these two attractors. The
gradient paths that connect these attractors with the critical
circle delineate the walls of the spindle-shaped correlation
cage.
This simple topology of I(r) persists at all values of the
internuclear separation R. As R increases, the magnitudes of
Intracule density
3403
I(r) at the critical circle and the cage point decrease in an
asymptotically exponential fashion, whereas the intracule
density at the attractors approaches (16␲ ) ⫺1 共Fig. 3兲. These
features of I(r) are readily explained with the help of the
‘‘Coulson–Fischer⫹r 12’’ wave function,23 which at the
H(1s)⫹H(1s) dissociation limit of the 1 1 ⌺ ⫹
g electronic
state gives rise to the two-electron density of
␳ 2 共 r1 ,r2 兲 ⫽N 兵 ␳ A 共 r1 兲 ␳ B 共 r2 兲 ⫹ ␳ B 共 r1 兲 ␳ A 共 r2 兲
⫹2 关 ␳ A 共 r1 兲 ␳ B 共 r1 兲 ␳ A 共 r2 兲 ␳ B 共 r2 兲兴 1/2其
⫻ 共 1⫹ ␤ r 12兲 ,
共6兲
where N is a normalization constant, ␳ A (r) and ␳ B (r) are the
one-electron densities pertaining to normalized 1s orbitals of
hydrogen centered at the nuclei A and B, respectively, and
␤˜0 as R˜⬁. Upon integration with respect to the extracule coordinate 共1/2兲 (r1 ⫹r2 ), such that ␳ 2 (r1 ,r2 ) yields I(r)
with maxima at r coinciding with the internuclear separation
vectors and a minimum at r⫽0. Indeed, inspection of Fig. 1
reveals that the positions of the attractors conform to the
approximate asymptotics of z attr⬇1.013 R⫺0.074.
This following of the positions of the internuclear separation vectors by the attractors is a manifestation of the
‘‘left–right’’ electron correlation. This longitudinal
FIG. 4. The correlation cage properties of the ground state of H2 vs R: 共a兲 V corr , 共b兲 N corr , 共c兲 W corr , and 共d兲 ␬.
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J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
Cioslowski et al.
correlation,24 which in the molecular orbital 共MO兲 theory is
accounted for by mixing of the (1 ␴ g ) 2 and (1 ␴ u ) 2 determinants, is of a nondynamical type. In contrast, the transversal
‘‘in–out’’ and angular correlation effects, described by the
admixtures of the (2 ␴ g ) 2 and (1 ␲ u ) 2 determinants24 are dynamical in nature. The magnitude of these effects is reflected
in the transversal extent of the correlation cage, i.e., the radius of the critical circle. Since this radius has the apparent
asymptotic dependence of r circle⬇0.104 R⫹0.499 on R, the
volume V corr of the correlation cage is proportional to the
third power of R at the dissociation limit 关Fig. 4共a兲兴. However, as I(0) becomes vanishingly small at the dissociation
limit, the magnitudes of N corr and W corr do not increase at
large R 关Figs. 4共b兲 and 4共c兲兴. The ratio ␬ 关Eq. 共3兲兴 falls off
with R, indicating the well-known inadequacy of the uncorrelated description of bond breaking in the ground state of
H2 .
In summary, the topology of I(r) in the 1 1 ⌺ ⫹
g ground
state of the hydrogen molecule is characterized by two entities, namely the attractors and the 共2,0兲 critical circle. The
positions of the former entities reflect the long-range longitudinal nondynamical correlation and that of the latter is determined by the transversal dynamical correlation effects.
IV. MORPHOLOGY OF I „r… IN THE 2 1 ⌺ ⴙ
g EF EXCITED
STATE OF H2
The energy of the 2 1 ⌺ ⫹
g state of H2 possesses two
minima with respect to R. According to the MO theory, these
two minima arise from the avoided crossing of two diabatic
states E and F.25 The united atom configuration of 1 S(1s2s)
becomes (1s ␴ g 2s ␴ g ) upon the separation of nuclei. In turn,
this configuration, which dominates around R⫽2 关 a.u.兴 , is
replaced by (2p ␴ u ) 2 关with significant contributions from
(1s ␴ g 2s ␴ g ) and (1s ␴ g ) 2 兴 at R⬎3 关 a.u.兴 . Finally, the EF
state dissociates into H(1s)⫹H(2s). The conflict between
topological features pertaining to these electronic configurations accounts for the complex morphology of I(r), which
involves not fewer than 11 distinct patterns of critical entities
共referred to as the sets I–XI in the following text兲 that are
interrelated by 7 cusp and three fold catastrophes 共Figs. 5
and 6, Table I兲.26
At the united atom limit, I(r) exhibits two radial
maxima at r⫽0.1626 and 3.3178 关a.u.兴 共set I, Fig. 5兲. In the
language of MO, the outer 共1,⫺1兲 critical sphere describes
an electron pair with each particle occupying one of the 1s
and 2s orbitals of the helium atom. The inner 共1,⫺1兲 critical
sphere, which delineates the correlation cage, stems from
electron pairs with both particles occupying the same orbitals. The maximum at rⴝ0 that would arise from these pairs
in absence of electron correlation is turned into the correlation cage by the electron–electron coalescence cusp. The radial maxima are interleaved by a 共1,1兲 critical sphere with a
radius of 1.7724 关a.u.兴.
As in the case of the ground state, separation of nuclei
results in 共⫹兲-cusp catastrophes that turn the 共1,⫺1兲 critical
spheres into pairs A and C of apical attractors 关Figs. 5共a兲 and
6共a兲兴 and the equatorial 共2,0兲 critical circles D and F 关Figs.
5共b兲 and 6共a兲兴. By the same token, the 共1,1兲 critical sphere
evolves into the pair B of bond points and the 共2,2兲 critical
FIG. 5. The evolution with R of the positions of the critical entities in the
electron intracule density of the EF excited state of H2 : 共a兲 critical points,
共b兲 critical circles. The broken lines denote the positions of critical circles
located off the z⫽0 plane.
circle E. The correlation cage is demarcated by the surface
AD, whereas the surface BE separates the basins of the attractors A and C. Another basin boundary is provided by the
surface that encompasses all of the critical circles. This topology, characterized by set II of seven critical points and
three critical circles, persists up to R⫽1.741 关 a.u.兴 . At this
value of R, a pair of 共⫹兲-cusp catastrophes spawns the
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Intracule density
3405
FIG. 6. The symmetry-unique sets of critical entities and the connecting gradient paths in the electron intracule density of the EF excited state of H2 共oriented
along the z axis兲 at R 关a.u.兴: 共a兲 1.5, 共b兲 1.8, 共c兲 2.4, 共d兲 2.75, 共e兲 2.85, 共f兲 3.0, 共g兲 3.15, 共h兲 4.0, 共i兲 4.8, and 共j兲 5.0. The hollow arrows indicate the displacements
of the critical entities upon the increase in R.
共2,⫺2兲 critical circle H from each attractor C, turning it into
a ring point G. These catastrophes do not affect the correlation cage, which grows in size with R. There are seven points
and five circles in this set III of critical entities 关Fig. 6共b兲兴.
As the H–H bond lengthens, the critical circles H travel
toward the equator. At R⫽1.878 关 a.u.兴 , they merge with the
共2,0兲 critical circle F, transforming it into the 共2,⫺2兲 critical
circle I. Again, the correlation cage, which becomes progressively elongated, is not affected by this 共⫺兲-cusp catastrophe. The resulting set IV of critical entities comprises seven
points and three circles 关Fig. 6共c兲兴. At R⫽2.477 关 a.u.兴 , another pair of 共⫹兲-cusp catastrophes occurs. The bond points
B turn into the cage points J, releasing the pair K of 共2,0兲
critical circles in the process. The new set V possesses seven
critical points and five critical circles 关Fig. 6共d兲兴. Again, the
spindle-shaped correlation cage does not undergo any discontinuous changes.
Signaling the end of the predominance of the
(1s ␴ g 2s ␴ g ) electronic configuration, the overall topology of
three concentric surfaces AD, EJ, and IG collapses at R
⫽2.805 关 a.u.兴 . By virtue of a pair of 共⫺兲-fold catastrophes,
each of the ring points G is annihilated by the cage point J,
producing set VI of three critical points and five critical
circles 关Fig. 6共e兲兴. The correlation cage remains intact.
With increasing R, the critical circles D and E gravitate
one toward another. Their mutual annihilation through a 共⫺兲fold catastrophe, which occurs at R⫽2.886 关 a.u.兴 , destroys
the boundary AD of the correlation cage, which is now replaced by the surface AI. Consequently, the cage expands in
a discontinuous manner 关Fig. 7共a兲兴. Set VII of the critical
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J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
FIG. 6. 共Continued.兲
entities consists of only six elements, namely three points
and three circles 关Fig. 6共f兲兴.
At R⫽3.138 关 a.u.兴 , two pairs of critical points emerge
from a pair of 共⫹兲-fold catastrophes. The ring points L provide a new boundary LI for the correlation cage, further reducing its already decreased volume 关Fig. 7共a兲兴. These
points, together with the cage points M, the attractors A, the
critical circle I, and the cage point at the origin constitute set
VIII of critical entities 关Fig. 6共g兲兴. The transition to the topology of I(r) pertinent to the (2p ␴ u ) 2 configuration is
completed at R⫽3.164 关 a.u.兴 . At this value of R, the critical
circles K merge with the cage points M through a pair of
共⫺兲-cusp catastrophes, yielding the pair N of bond points.
The resulting set IX consists of seven critical points and one
critical circle 关Fig. 6共h兲兴. The correlation cage is unaffected
by this transition.
Further stretching of the H–H bond brings about only
minor changes in the topology of I(r). At R⫽4.772 关 a.u.兴 ,
the 共2,⫺2兲 critical circle I spawns the pair Q of 共2,⫺2兲
circles, consequently becoming the 共2,0兲 circle O. Set X,
which is produced by this 共⫹兲-cusp catastrophe, has as its
elements seven critical points and three critical circles 关Fig.
6共i兲兴. The circles Q travel toward the poles of the correlation
cage, merging with the ring points L and producing a pair of
attractors P at R⫽4.895 关 a.u.兴 . Continuity in the properties
of the correlation cage is preserved in both cases.
The final set XI of topological entities consists of two
pairs A and P of attractors, the pair N of bond points, the
cage point at the origin, and the 共2,0兲 critical circle O 关Fig.
6共j兲兴. The gradient paths OP trace the walls of the correlation
cage. The surfaces that pass through the bond points N and
the plane that encompasses the circle O constitute the boundaries of the attractor basins. The positions of the attractors A
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J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
Intracule density
3407
FIG. 6. 共Continued.兲
and the bond points N, which are given by the approximate
asymptotic formulas z A ⬇1.040 R⫹0.211 and z N ⬇1.052 R
⫺1.453, respectively, follow the internuclear separation vectors. In contrast, the size of the correlation cage appears to
stabilize with increasing R, the positions of the attractors P
conforming to the apparent asymptotics of z P ⬇1.108
⫺23.55 R ⫺3 and the radius of the critical circle O amounting
to 0.867⫹0.008 R. These asymptotics are consistent with the
H(1s)⫹H(2s) dissociation limit corresponding to I(r) with
primary maxima 共the attractors A兲 that arise from pairs of
electrons residing on different nuclei. The secondary maxima
in such I(r) 共the attractors P兲 are accounted for by electron
pairs located near the bond midpoint with each particle occupying one of the 1s and 2s orbitals.
The properties of the correlation cage are directly related
to its shape and spatial extent. Inspection of Figs. 7共a兲–7共c兲
that display the dependencies of V corr , N corr , and W corr on R
reveals a common pattern of an essentially continuous evo-
lution with the internuclear separation that is disrupted by
two of the aforedescribed fold catastrophes. The catastrophe
at R⫽2.886 关 a.u.兴 brings about a tremendous increase in
V corr 关Fig. 7共a兲兴, which is accompanied by sudden rises in the
number of strongly correlated electron pairs N corr 关Fig. 7共b兲兴
and their contribution to the electron–electron repulsion energy W corr 关Fig. 7共c兲兴. The values of these properties quickly
decrease with R, falling back to their previous levels prior to
the second fold catastrophe at R⫽3.138 关 a.u.兴 . Consequently, the anomalous behavior of the correlation cage is
confined to an interval of R that spans less than 0.3 关a.u.兴.
Since this interval coincides with the region of the avoided
crossing between the diabatic E and F states 共the intermediate energy maximum, Fig. 8兲, the abrupt variations in the
shape and the spatial extent of the correlation cage are
readily attributable to the rapid changes in the character of
the underlying electronic wave function. This observation is
further corroborated by the low values of the ratio ␬ in this
TABLE I. Catastrophes that interrelate distinct topologies of I(r) in the 2 1 ⌺ ⫹
g state of the hydrogen molecule
at nonzero internuclear separations.a
Catastrophe
Catastrophe locationb
R
Type
Effectd
x
z
1.741
1.878
2.477
2.805
2.886
3.138
3.164
4.772
4.895
共⫹兲-cusp
共⫺兲-cusp
共⫹兲-cusp
共⫺兲-fold
共⫺兲-fold
共⫹兲-fold
共⫺兲-cusp
共⫹兲-cusp
共⫺兲-cusp
C˜G⫹H
F⫹H˜I
B˜J⫹K
G⫹J˜ ␾
D⫹E˜ ␾
␾ ˜L⫹M
K⫹M ˜N
I˜O⫹Q
L⫹Q˜ P
0.000
4.591
0.000
0.000
1.089
0.000
0.000
0.906
0.000
⫾4.966
0.000
⫾4.798
⫾6.173
0.000
⫾0.807
⫾1.118
0.000
⫾0.906
I(r) c
5.11 ⫻
4.85 ⫻
3.14 ⫻
2.37 ⫻
1.88 ⫻
7.19 ⫻
8.54 ⫻
5.27 ⫻
5.21 ⫻
10⫺4
10⫺4
10⫺4
10⫺4
10⫺4
10⫺4
10⫺4
10⫺3
10⫺3
All quantities in 关a.u.兴. The molecule is oriented along the z axis.
Coordinates of the critical entities involved in the catastrophe.
c
I(r) at the critical entities involved in the catastrophe.
d
See Figs. 5 and 6 for the letters denoting individual critical entities. The symbol ␾ stands for an empty set.
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a
b
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Cioslowski et al.
J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
FIG. 7. The correlation cage properties of the EF excited state of H2 vs R: 共a兲 V corr , 共b兲 N corr , 共c兲 W corr , and 共d兲 ␬. The broken vertical lines indicate the
values of R at which the catastrophes in I(r) occur.
region 关Fig. 7共d兲兴 that are indicative of relatively weak local
correlation effects, i.e., the predominance of nondynamical
correlation stemming from a multiconfigurational nature of
the wave function.3
present study confirms the association between the low values of this ratio and the relative weakness of local correlation, i.e., the predominance of nondynamical correlation
effects.3
V. DISCUSSION AND CONCLUSIONS
The differences in electronic structures of two 1 ⌺ ⫹
g
states of the hydrogen molecule are vividly reflected in their
intracule densities I(r). The ground state, in which two electronic configurations predominate over the entire range of
internuclear separations, is associated with two distinct topologies of I(r). On the other hand, no fewer than 11 unequivalent sets of critical entities that arise from conflicts
between topological features of I(r) pertinent to different
electronic configurations are found for the EF state.
An occurrence of a catastrophe that interrelates individual topologies of I(r) constitutes a very sensitive indicator of a change in the electronic structure of a given molecular system. When a range of molecular geometries where a
single electronic configuration does not longer dominate is
approached, an abrupt increase in the volume of the correlation cage is observed. This conspicuous change is accompanied by a concomitant drop in the magnitude of ␬. The
FIG. 8. The energy of the EF excited state of H2 vs R. The broken vertical
lines indicate the values of R at which the catastrophes in I(r) occur.
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J. Chem. Phys., Vol. 111, No. 8, 22 August 1999
The data presented in this paper also demonstrate the
usefulness of topological analysis of I(r) in the detection and
characterization of various types of electron correlation in
atoms and molecules. A complete fingerprint of electron correlation is provided by the topology of I(r). Detailed insights into the relative relevance of dynamical and nondynamical correlation are easily gained from analysis of the
shape and the spatial extent of the correlation cage. Although
the topological features of I(r) in systems with more electrons are expected to exceed those uncovered here in intricacy and complexity, they are certain to lend themselves to
rigorous interpretation.
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under Grant No. CHE-9632706 and by KBN under
Grants Nos. T11F00712 and SPUB/COST/D9.
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For an animated version of Fig. 6 see the ‘‘Research in Pictures’’ section
of the senior author’s homepage.
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