The Influence of Electrons on Each Other in a Molecule: Correlation of Electron Motions in H2 Robert C. Dunbar Case Western Reserve University. Cleveland, OH 44106 In its motion in a molecule, an electron responds in several ways to the motion of other electrons. We can think in terms of three kinds of effects: (1) The electron moves through the molecule under the influence of the charges of the nuclei and also the cloud of charge created by the time-averaged positions of the other electrons. (2) Because of Coulomb repulsion between pairs of electrons, the electron tends to avoid the instantaneous position of eachof the other electrons. (3) The Pauli principle of exchange antisymmetry of the wave function results in the electron avoiding (in some cases) or favoring (in other cases) the instantaneous positionsof other electrons. Most of us learn about these effects without having much concrete feeling for how they work and how important each of them is in governing the resulting electron distributions in , a molecule. This paper aims t o give a pictorial description and comparison of these electron motions in the simplest interesting case, which is the Hz molecule. Effect 1 is familiar. and is incor~oratedinto all realistic descriptions oielectr"n distributions. Most interelectron effects. such as "shieldine" effects, can be accounted for in a fairly satisfactory way-by this approximation. In a good molecular orbital description of a molecule, each of the molecular orbitals that an electron can occupy takes into account the average distributions of the other electrons in this way. Effect 2 is what is usually referred t o as the "electron correlation". The attempt to describe the correlated electron motion accurately motivates much current research in quantum chemistry. Effect 3 also has the result of giving major correlation of the instantaneous positions of electrons in the molecule, as we will see below, hut i t is not considered part of the "correlation problem", since i t comes about automatically if a properly antisymmetric wave function is written. Any wave function that is antisymmetrized by the technique of writing i t as a Slater determinant automatically correlates the electron motions in this way. Hz is clearly the simplest molecule in which we can look a t the results of electron-electron interactions; it is nevertheless interesting because i t can serve as a useful model for the behavior of the pair of electrons in a two-electron bond in a complex molecule. Early in the development of molecular quantum mechanics, accurate wave functions were written for several Hz quantum states in terms of 10-12 cleverly chosen basis functions, which we can use to explore these questions (1-3). We are going t o study the effect of the electrons on each other's motion by posing the question: If electron 1is known to be a t a particular, fixed place in the molecule, how does electron 2 distrihute itself through the molecule? We will use the description of electron 2 contained in the wave function rIr(rl,rz), which is the amplitude for finding electron 1a t r, and electron 2 a t rz. The probability density for finding If we want to electron 1 a t rl and electron 2 a t r 2 is 1q2. specify that electron 1is at roand ask for the probability of finding electron 2 a t rz,we want to look a t the function
This is a function of one electron position (rz), which describes the probability distrihution of electron 2 in space, given that electron 1is always a t ro. The pictures shown here were generated from the wave functions by setting rl to the chosenvalue ro, and writing the probahility distrihution function for electron 2 as in eq 1. Then a randomizing procedure was used to place dots on the plane, according to the rule that the probability of a dot appearing a t a given location on the picture is equal t o the probahility density for finding electron 2 a t that point. The pictures thus show directly the probahility (or charge) distribution of electron 2. In the research literature. a recent illustration of displaying correlated wave functions by freezine one of the electrons and m a.. ~ o i n the e other was ~ r o v i d e d h i Krause and Berry in an extensive study of the wave functions of some alkaline earth atoms (4). We will restrict our displays to the case where both electron 1and electron 2 are in a plane which contains the two nuclei. It does not matter whic-h such plane we choose, since the 2 functions considered here have cylindrical symmetry. The restriction t o the plane containing the nuclei might seem severe, but actually the essential features appear pretty much the same if rois above or below the rzcross-sectional plane displayed, or if the displayed rz plane is displaced from the nlane of the nuclei (but still narallel to the H-H axis). Since a comprehensive display of even these "small" six-dimensional wave functions would reauire far more ~ i c tures than we can show here, we will talk ahout the electron correlation effects within the confines of this in-plane restriction, without missing any important features.
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Ground State
First we will look a t the X12, ground-state wave function of Hz. Since the two electrons are paired in a closed shell, there is no electron correlation due to the antisymmetry of the function, that is t o say, effect 3 is absent. Thus this state will let us examine the roles of effects 1and 2 alone. (All the ground-state calculations use the known molecular H-H distance of 0.74 A.) Figure 1: Independent electrons. In the absence of any electron-electron interactions a t all, the wave function is the product of two ground-state, one-electron wave functions for the HA7molecule (5). Klectron 2does not care where ele(.tnm 1 is located if the two electrons are independent, so the electron 2 probability distrihution function in this approximation for Hz is the same as Hz+,as shown in the figure. Figure 2: Hartree-Fock function. T h e Hartree-Fock ground state function (6) is the approximate function in which effects 1 and 3 play their full roles, but effect 2 is absent. That is. electron 2 is distributed accordine t o a Schrodinger equntion containing the nuclear powntials plus the Coulomb reoulsion ~ o t e n t i aol i a cloud of charge corresponding to thetime-averaged position of electron 7. Again, we only need one picture t o display this, since we need not specify a position for electron 1. The figure shows the elec-
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Number 6 June 1989
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tron 2 distribution in this approximation. We would expect this function to be alittle more spread out than Figure 1,due tonuclear screening by electron 1,but as we see, this effect is small, and there is very little difference between these two functions. Figure 3a, h, and c: Exact wave function (1).Using the exact wave function, we can now specify various positions of electron 1 and look at how the electron 2 distribution changes. The figures show three such choices, the position of electron 1being shown by the 0 symbol. We can see that the correlation (effect 2) is substantial, with electron 2 avoiding the vicinity of electron 1 strongly. However, almost all the correlation is along the H-H direction: When electron 1is above the plane of the proton (Fig. 3a), electron 2 is pushed helow this plane only to a minor extent, but when electron 1 is at one end of the molecule, electron 2 is pushed very strongly to the other end (as in Figs. 3b and 3c).
symmetrizing this function with a singlet spin function to give a singlet state, while the 39state is the triplet state arising by antisymmetrizing the same molecular orbital state with a triplet spin function. Thus these two states are similar in that they are derived from the same molecular orbital configuration, and comparing them at similar H-H distances' gives a fairly direct view of the operation of effect 3 in singlet and triplet states. Tri~letState, %. In a triplet state, the two electronr have the same quantum numbers, so that according to the Pauli exclusion winciple they cannot occupy the same point in space.
'
The '2, state is weakly bound, with H-H distance 1.28 A. Theg2. state is repulsive, and of the several H-H distances for which wave functions weregiven in ref 2. 1.22 A waschosen for the present plots.
ExcIted States
We need to look at excited states in order to get a view of the effect of antisymmetrizing the wave function (effect 3). In order to illustrate this effect, we have chosen a pair of excited states designated B12, and b32.. These two states are hoth derived from the molecular-orbital excited state in which one electron remains in the a, lowest bonding molecular orbital, while the other electron is excited into the a,* antibonding molecular orbital. The '9state arises by anti-
GROUND STATE I
ION GROUND STATE
I, -1.6
,
, -1.0
,
,. ., .. , . ., -0.5
0.0
POSITION
, 0.6
,
;., 1.0
, I 1.5
(A)
F g v e 1 Plot of the probaoili distrioution for an elemon in the 0. abnal of H,'. The positions of lhe two protons are indicated oy the rymoois.
HARTREE-FOCK
I ;, -1.5
, -1.0
. ., . .,.. -0.5
,
, 0.0
POSITION
; , 0.5
,
, 10
, I
(A)
POSITION (A)
Flgwe 2. The probability disbibution f a one of the electrons in the HZground slate in the HanregFock approximation.
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Figure 3. Robabilny distribution(eq 1) for electron 2 in the ground state of Hs. given mat elecbon 1 is at the posfiion designated by the 0 symbol. (Plotled from me accurate wave tunnion of ref 1.)
TRIPLET STATE (INDEPENDENT ELECTRONS)
Figure 4a, b, and c: Exact wave function (2). The spatial distribution of electron 2 in the 32,. state is shown in Figure 4 for three locations of electron 1. Electron 2 always has zero probability of being a t the same place, as expected, and in fact the antisymmetrization of the wave function leads to a nodal surface (a surface over which the electron 2 probability density is zero) that always passes through electron 1.The extent of correlation of the motions of the two electrons introduced by effect 3 is enormous and is far greater than the other two effects. Figure 5: Independent electrons. It is easy to isolate the influence of effect 3 in the exact wave function by comparing it with the antisvmmetrized wave function for totallv independent electrons (no coulomb repulsion). This independent-electron function is constructed from the a, and a,*
. .
... -2.0
0.0
-1.0
POSlTlON
TRIPLET STATE
I
I
Figure 5. Probability disulbutim far eiecuon 2 as in Figure 4, but uslng the apprnxlrnate Independent-electron wave function of eq 2. The Internuclear distance and lhe electron 1position chosen correspond to Flgure 4b. SINGLET STATE
.
1
I,
, -2.0
,
', -1.0
.,
,
,
0.0
, 1.0
,
,
2.0
1.0
(A)
,.
,I
2.0
POSITION (A)
1,
, -3.0
Fiaure 4. Probabilitv distribution for electron 2 in the 3Z, excited state Of K. g ven mat e actran 1 0s at me pole on desgnated by The 0 symbol iP oned from the accurate wave tunct8on ol ref 2 1 The wave functlon ltself changes ~ognauass me nodal surface n each case, almougn lhe prooaoll~ryaostrmtian is always nonnegative
,
,', -1.0
, -LO
,
, 0.0
.
I
1.0
1
.
2.0
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10
POSITION ( A ) Figure 6. Probabilitydlstrloulon faelectron 2 in me '.X,exclted state (Plonen from th. accurate wave function of ref 3 ) Agam the wave fmalon changes sign across the nodal surface.
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465
SINGLET STATE
tions of electron 1as above. Again the Pauli-principle correlation is overwhelming, with electron 2 spending most of its time on the same side of the molecule as electron 1. Figure 7: Independent electrons. Just as with the triplet state, we canseparate out the part of the electron correlation due to antisymmetrization by comparing the exact wave function with the independent-electron function in which the Coulomb repulsion is zero. For this case, the independent-electron function takes the form
(INDEPENDENT ELECTRONS)
'*
-9.0
-2.0
-1.0
0.0
POSITION
LO
2.0
3.0
(A)
molecular orbitals (4) of Hz+ at the same H-H separation (1.3 A), as follows: = ag(rl)c,,*(r2)- + J o U * ( r l )
(2)
This function is plotted for one position of rl in Figure 5 and can be compared directly with the corresponding exactfunction in Figure 4b. We see that the antisymmetrization accounts for most of the tendency of electron 2 to avoid the position of electron 1.The Coulomb repulsion acting in Figure 4b pushes the electron 2 densitv even more to the left than its-distribution in Figure 5, butthis Coulomb repulsion correlation is only a rather small part of the overall correlation. Singlet State, '2.
In the singlet state the spin function is antisymmetric, and the space part of the wave-functionis symmet& for electron exchange. The two electrons can thus occupy the same point in space, and we will see that they actual1;favor beingclose together in this state. Figure 6a, b, and c. Exact wave function (3).The electron 2 distribution is shown in the figures for the same three posi-
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Journal of Chemical Education
+ aJr2)au*(rI)
(3)
This function is plotted for one position of r, in Figure 7 and should be compared directly with the exact function in Figure 6b. Again it is clear that the Coulomb repulsion (effects 1 and 2) has only a minor influence in comparison with the Pauli-principle correlation (effect 3). but that the electron repulsion does push electron 2 somewhat more to the left side of the molecule, away from electron 1.
Figure indenen- 7. PrDabllii distribution for elechon 2 usintl" Me amroxlmate .. dent-elechon wave functionof eq 3. me imsrnLeear Ulotance and me electron 1 posnlon cnoyln comespond to Flgure 6b.
3*
= uJrl)a,,*(r2)
Conclusion We have looked at three types of correlation of the motion of the interacting electrons in Hz: Effect 1,average Coulomb repulsion; effect 2, instantaneous Coulomb repulsion; and effect 3, Pauli-principle correlation. In the ground state, where effect 3 is inoperative, effect 1is found to give little chanee relative to the iudeoendent-electron a~~roximation. but &e correlation due to effect 2 is substan&l. In the two excited states, the correlation of electron positions due to effect 3 is very great, with the two electrons strongly avoidine each other in the t r i ~ l estate t and tendinn to be near each other in the ringlet stace. The correlation inihese states due toeffect 2 is smaller, but still noticeable by comparison with the independent-elktron approximation:
Acknowledgment We are grateful to John Hays for his help in making the plots and to the Frank Hovorka Computer Center in the Chemistry Department of Case Western Reserve University for providing the means to compute and plot the pictures. Literature Clted 1. James, H. M.; Caa1idge.A S. 3. C k m . Phya. 1935.1.823. 2. Cwlidre, A. S.:James,H. M. J. Chem. Phva. 1938.6.730. 3. rmmi. R.0. J (.hem. rhya I Y X S , ~ . I T ~ 4. Krnu%.l. I..&rrv. K S J Chsm P h ~ a198583.3123 5 Hava I) K ;Ledaham. K.: Supan.A L. Phrl Tram. Ro) 6 (:nuIwn. (' A P r o , r'omhnda~Phrl Snr 1938, U .204
