Electronic Confinement of Molecules in Microscopic Pores. A New

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J. Phys. Chem. 1994,98, 10863-10870

10863

Electronic Confinement of Molecules in Microscopic Pores. A New Concept Which Contributes To Explain the Catalytic Activity of Zeolites C. M. Zicovich-Wilson and A. Corma* Instituto de Tecnologia Quimica, Universidad Politknica de Valencia- C.S.I.C., Av. Naranjos, s/n-46022 Valencia, Spain

P. Viruela Departament de Quimica-Fisica, Universitat de ValPncia, Dr. Moliner 50, Burjassot (ValPncia), Spain Received: April 26, 1994; In Final Form: August 7, 1994@

The chemical consequences of the spatial confinement of electrons when an ethylene molecule is trapped into a microscopic cavity was studied. The JC electronic system of ethylene was modeled by means of the Huckel Molecular Orbital Theory in order to provide an easy interpretation of the mathematical results. It was determined that the reactivity of molecules was increased in two ways: An intrinsic one, which accounts for the weaking of the bond by decreasing the n-n*energy gap; and in an extrinsic way, which is related to the increase in the energy levels of the guest molecule with respect to those of the catalysts acid sites. This results in an enhancement of the base character of the molecule owing to the confinement. Finally the order of magnitude of the increment of energy produced on the ethylene n system when it is confined in cavities of different pore sizes was determined. From our results it is expected that, for cavities smaller than 5-6-A size, the orbital energy levels of the guest molecule are increased by an amount of around 1 eV or greater.

1. Introduction It is generally accepted that the structure of channels and cavities with molecular sizes in zeolites have a strong influence on diffusion and solvation effects of reactant molecules, and they can become determinant for the catalytic activity of these materials. Diffusion, which involves the dynamics of the molecule into the channels, is described by a statistic mechanical model which correlates well with experimental results.’ Solvation or cavity effects that involve physicochemical interactions between the zeolite pore and the guest molecule are not well quantified and are still a matter of discussion.2 The type of interactions normally considered are the following. Coulombic Effects. These are produced by the charge distribution along the framework owing to the partial ionic character of the aluminosilicate crystals. This charge distribution generates a strong Coulombic field into the cavities, which might influence the chemical behavior of the guest molecules.2a Indeed, in zeolites, because of the zeolite geometry, i.e., cages, channels, side pockets, etc., and the charge distribution, energy field gradients will exist and they can activate the reactants when being in the confined situation.2b,c In this case, the field and its gradient are very strong in the proximity of the atoms at the walls and vary rapidly over 1 or 2 A.2d Coordination Effects. These are produced by a Lewis acidbase type interaction among the guest molecules and certain sites of the framework in the cavity. In other words, the zeolite catalyst can present electron acceptor-donor centers, which favors the adsorption of the molecules.2e Weak electron interactions account for forces of the van der Waals type, which could produce the “docking” of a molecule into zeolite cavity.2f In this sense, it has been proposed that sorbate molecules in microporous solids, such as zeolites, tend to optimize their van der Waals interaction with the surroundings.2f*gThese van der Waals interactions are amplified

* To whom correspondence should be addressed. @

Abstract published in Advance ACS Abstracts, September 15, 1994.

0022-365419412098-10863$04.50/0

by the surface curvature of the pore walls, which interact with the sorbed molecules. This was called the confinement effect. In this work, we present a new concept which can contribute to explain the high reactivity and acidity shown by zeolites when bridging hydroxyl groups interact with hydrocarbons in the cavities and also when the catalyst pore acts as a microreactor where reactions occur. This new concept that we have called electronic confinement or “boxing effect” will become specially important when the size of the guest molecules matches the size of the cavities. Since, owing to the partial covalent character of the aluminosilicate crystal^,^ electrons are not localized on the framework atoms, but they are partially delocalized through the bulk, this causes the density of the orbitals, (Le., the probability for finding the electrons) of the guest molecule to suddenly drop to nearly zero when reaching the walls of the zeolite as a consequence of the short-range repulsion with the delocalized electronic clouds of the lattice. This implies that a contraction of the orbitals of the guest molecule will occur, with the consequent changes in its energy levels. It can be easily understood that this effect should be more important when the sizes of the zeolite cavity and the guest molecule are closer. The implications of the electronic confinement effect on chemical reactivity are obvious if one considers that changes in the energy levels of the guest molecule could imply a preactivation of the molecule when residing in the pore or zeolite cavity. In previous works, this effect has been investigated from the point of view of studying hydrogenic impurity states in the confinement produced by quantum dots, quantum wires, and other low-dimensional semiconductor structures! Very recently, a few interesting theoretical works have been devoted also to study semi-infinite (one-side) confinement, as a simulation of the influence of the presence of a surface on the properties of a hydrogen atom and a hydrogen m o l e c ~ l e . ~ Here we present an approximation based on the Huckel molecular orbital theory to show in a qualitative way the changes which might occur on the orbital energy levels of 0 1994 American Chemical Society

10864 J. Phys. Chem., Vol. 98, No. 42, 1994

Figure 1. Macroscopic particle in a potential wall: (a) when being unconfined; (b) when being confined. a molecule such as ethylene and, by extension, other x systems when located in a microscopic cavity in which the electronic confinement is produced. The implications of this effect on catalysis in zeolites are also discussed. The paper is organized as follows: Section 2 is a general discussion about the physical meaning of spatial confinement in microscopic mechanic systems. Sections 3 and 4 contain a brief description of the model and theoretical method used, which is developed in detail in Appendixes I and 11. In section 5 the obtained results are specifically discussed. And finally in section 6 the catalytic implications of the electronic confinement of reactant molecules into the zeolite pores or cavities is discussed.

2. Electronic Confinement When a macroscopic particle is located in a one-dimensional potential well which has an energy minimum at a given point ( X O ) of the space (Figure la), it becomes evident that, when the particle is trapped into the potential well, its motion tends to reach the equilibrium position in an oscillating way. When energy is given to the system, the amplitude of the oscillation will become larger until the particle will fall out of the well. Let us now suppose that we confine the system by adding two vertical infinite walls around the equilibrium position xo (see Figure lb). Thus, it is expected that in the lowest energy states, i.e., when the oscillations do not reach the walls, the dynamics of the system should not change with respect to the unconfined one. In this case, only the highest energy states will be modified starting from the energy sufficient to allow the particle to strike the walls. The situation becomes different for microscopic systems which are described not by classical but by quantum mechanics. In this case the energy levels are quantized, and a probability function should be used to define the position of the particle at a given energy state. It can be expected that the probability for the particle reaching the walls will not be zero even if the system is in the lowest energy state (Figure 2). Thus, for the microscopic system, not only the higher but all the energy states should be affected by the confinement, since every one has some kind of interaction with the enclosure walls. This interaction

Zicovich-Wilson et al.

Figure 2. Microscopic particle in a potential well: (a) when being unconfined; (b) when being confined. The dashed curve depicts the probability function of the particle.

t

t

Figure 3. Change of the energy levels of the particle when confining the microscopic particle. On the left-hand side the energy levels for the unconfined system ace depicted and on the right-hand side, the corresponding levels for the confined one. should increase the energy levels of the system to an extent which depends nearly on the probability of collision with the walls. In this way it can be expected that the more diffuse, i.e., the less localized around the potential minimum position, the probability function of a given state in the unconfined system is, the higher the number of collisions with the walls per unit time will be when confining the system and, in him, the more energetic the corresponding confined state will be (Figure 3). It is clear that the above model is an ideal one, since in real situations the potential walls are neither vertical nor infinite. Nevertheless the ideal model can still be used as a first approximation to describe some real cases. Let us consider the case of a guest molecule inside a zeolite cavity. Since quantum chemical calculations3have proven that Si-0 bonds in zeolite have a clear covalent character, it can be considered that the valence electrons are not localized around the anions as happens in the purely ionic crystals, but they are distributed all over the framework atoms as a partially delocalized electronic cloud. If this is so, at relatively short distances (2-3 A) between the guest molecule and the walls of the zeolite cavities the electron-electron repulsions between them will start to be important. Owing to the electronic delocalization inside the zeolite bulk, this electron-electron repulsion impedes the electrons of the guest molecule to easy penetration through the walls of the zeolite cavity. It then appears that this situation can be simulated by a model in which the potential produced by the electron cloud of the zeolite walls is substituted by a vertical infinite potential wall. It should be realized that here

Electronic Confinement of Molecules

J. Phys. Chem., Vol. 98, No. 42. 1994 10865 (except for a normalization factor) the bonding and antibonding

x molecular orbitals, and txnn. are the corresponding energies

fW> y

~-

Id)= la’)

+ Ib’)

td = a’

+

Ihl

~~

Figure 4. Model for the ethylene molecule n system: (a) confined system; (b) unconfined system. In both cases the n orbitals are

qualitatively depicted. the situation is essentially the same as in the simpler case described above, concerning a particle trapped in a onedimensional potential well. The particle (the electron in a oneparticle quantum chemical model) is now moving around the nuclei because of the electrostatic attraction forces, in the same way that in the one-dimensional panicle model it moved around xo (see Figure I). However. it is also necessary to take into consideration that in the latter, being a three-dimensional system, the spacial distribution of the electrons of the guest molecule and its relation with the cavity shape should be also considered. In the next section we will do this analysis by means of the HMO theory in order to study the electronic confinement effect on an ethylene molecule. This should allow us to extract qualitative conclusions about the expected changes in x systems (alkenes, polyenes, aromatics. etc.) due to the electronic confinement effects generated when introducing such molecules into the zeolite cavities.

3. HMO Treatment of the Ethylene K Electrons Starting from the closed-shell single-determinantal approximation which treats each electron as moving independently in a mean electrostatic field of the other electrons and the nuclei, HMO theory is a method that allows one to obtain a chemical characterization of z systems without need of involved calculations. Its goal is to give an adequate description of II molecular orbitals (their energy and atomic orbital composition) by only using two empirical parameters. namely, the Coulomb integral a and the resonance integral p. The resulting model is oversimplified to he used for accurate calculations. But, in change, it is conceptually transparent. This transparency is the main fact that motives its use in this paper. In the case of ethylene where there are only two electrons in the II system, the solutions given by the HMO theory are

In)=la)+Ib)

given as a function of a and B parameters. A confined system model can be built by locating the ethylene molecule into a cavity which, for simplicity, may be simulated by two parallel planes at a distance equal to the diameter of the cavity. The ethylene would be located equidistant (at d distance of each plane) and parallel to the border planes. The election of this model does not results in a serious loss of generality, since the essential features present when confining the electrons of a molecule into a zeolite cavity are present here. The symmetry of the system does not change with respect to the free molecule, and this implies the solutions will conserve the same form as for the unconfined system:

t,=a+B

where la’) and Ib’) are the 2p, orbitals of the confined carbon atoms, d and d*the corresponding bonding and antibonding molecular orbital, and a’and are respectively the Coulombic and resonance integrals for the confined system. The atomic orbitals la’) and Ib’) are not m e p orbitals. They have as the normal p orbitals a node at the molecule plane but they are also adapted to the boxing, which makes them vanish beyond the limit of the cavity. This comes from the condition that the probability of finding an electron belonging to the molecule outside the cavity should be null (see Figure 2b). In this way, one properly models the difficulty of the electrons of the guest molecule (in its lowest energy states) to penetrate through the surface electron clouds.

4. The Evaluation of the Electron Confinement Effects for the Ethylene Molecule

To evaluate the energy changes produced by the electronic confinement we will make use of an expression which relates the energy variation of a given electronic state when confining the system, with the corresponding wave functions:

A€=--

hz s,dr WX)(%W(X) 2m L d r Y*(x)Y’(x)

where ‘P and ‘4’ are the wave functions of a given energy state of the isolated and confined system, respectively. In the numerator the integration is performed over S.the cavity border surface, and ahl indicates the directional derivative normal to the surface S at any point x E S. The integration in the denominator is over the cavity volume V. Equation 2 is completely general and it is developed in Appendix I. Nevertheless, in order to use eq 2 in an exact way. the wave functions corresponding to the confined system have to be known, something which is not generally the case. However, as is shown in Appendix 11, by taking into account the symmetry conditions imposed on the model (see section 3). the energy changes of the one-electron states of ethylene may be expressed as

At, = cd - t, = A a A+ = E+ where la) and lb) are the 2p, atomic orbitals on carbon atom a and b, respectively (see Figure 4). In)and In*)are respectively

(2)

- E,.

+ AB

= A a - Ap

where A a and AD are (in atomic units)

(3a) (3b)

10866 J. Phys. Chem., Vol. 98. No. 42. 1994

Aa=-2

Zicovich-Wilson et al.

dy a(x.y,d)ap’(x,y,d)

& dz L-hdy a(x.y.z)a’(x,y,z) d

(3c)

1-hdy a(x.y.d)a,b’(x.y,d)

Ab=--

(3d) [-&

dy a(x,y,z)a’(x,y,z)

In these expressions a(x,y,z) and b(xy.2) are the 2p, atomic orbitals on atom a and b respectively, while a’(x,y,z) and b’(x.y.2) are the equivalent atomic orbitals but for the confined molecule; 3, indicates the partial derivative with respect to z direction and d is the distance among the molecule and the cavity border. Despite the fact that we do not know the actual expression of the orbitals a‘(x.y.z) and b‘(x.y,z), we can state that Aa and Ab are always positive quantities. Since the topology and the symmetry of the confined and free models are in this case the same. one can expect that the orbitals of both models will have also the same symmetry and topology; Le., there should be a unique node plane at z = 0 (see Figure 4). Now let us see in Figure 5 a representation of both orbitals along the z axis, being x and y fixed at an arbitrary point, and suppose that the orbitals are normalized to yield fwa’ d r = 1. It is easy to notice that when a(x,y,z) is positive at z = d, then a’(x,y,z) decreases when increasing z, which means that its derivative in such a direction should be negative, and therefore the numerator of eq 3c will also be negative; whereas by hypothesis the integral in the denominator of eq 3c will be positive. This implies that Aa 2 0. In the same way one can obtain that AD 2 0 in eq 3d. This is because a’ and b’ are simultaneously increasing or decreasing at the cavity border. By combining eqs 3a, 3b. and 1‘. one obtains the following conditions for the Coulomb and resonance integrals of the confined system

a’= a + Aa 2 a

B’ = /9 +Ab t fi

(4)

This result, which has been obtained for the ethylene molecule, may be easily extended to any conjugated polyene system.

5. Discussion Since a and are both negative quantities. expression (4) indicates that the absolute values of a and 9, decrease when confining the molecule. In Figure 6 we have presented a qualitative correlation diagram of t h e n orbital energy spectrum of the ethylene molecule when it is confined to a cavity. One can see that, as expected, the confinement does produce an increase of all the energy levels, but the increase is different for the bonding and antibonding n orbitals: the bonding n orbital seems to be more “sensitive” to the electron confinement effect than the antibonding A* one. This may be explained by the fact that the n* orbital has one nodal plane which intersects the cavity border whereas the n orbital does not. At the intersection of the cavity border with the node, there should be a null electron density since the node corresponds to the zone where the wave function (and therefore the electron density) vanishes. Therefore, one expects that the collision frequency of the electron with the cavity surface will be smaller, and so will be the energy associated to the confinement. when a node crosses it. This is in agreement with a previous study of the hydrogen atom centered in a spherical cavity’ in which it was observed

Figure 5. Schematic representation of the rY function (see the text) for x and y fixed at any point of space. The thin line depicts the unconfined wave function and the thick line the confined one. Dashed line represents the cavity limit.

Figure 6. Qualitative description of the IT orbital energy spect” for the ethylene molecule. On the left-hand side the SpeCtNm of the unconfined molecule is depicted and on the right-hand side. the spectrum of the confined molecule. that the larger the angular momentum of the electron, the smaller the increment of energy produced by the confinement. It is well-known that the angular momentum of the atomic orbitals is related to the number of node planes which intersects the nucleus and. for instance, every surface which surrounds it.

6. Implications of the Electronic Confinement Effect on Catalysis As it is shown in Figure 6 the energy gap between n and n* orbitals decreases, when ethylene is confined, and, therefore, the n bond becomes weaker than when unconfined. Indeed, a small bonding-antibonding orbital energy gap means that a quasi-degeneracy of states should occur at the ground state, which affects the electronic structure of the bond. A single configuration model of the many-particle wave function cannot be assumed and. at the minimal basis set level (which is implicit in HMO theory), one expects that the actual wave function can be properly represented by a multireferential function which accounts for a partially biradical character of the bond.8 Then the smaller the n-nb gap is, the more the system will act as a molecule with unpaired electrons on each carbon atom. Thus, the system would become intrinsically more reactive when confining the molecule into the catalyst cavity. This means that reactions which involve a loss of the n character of the molecule would be favored. Consequently, addition reactions of alkenes or polyenes should occur more easily when performing the reactions in molecular sieves. Also reactions in which addition implies a cyclization in a concerted way, such as the DielsAlder type. are also expected to be favored in micropores of zeolites. Several experimental results on the catalytic reactivity of zeolites match with this analysis? The increase of reactivity produced by decreasing the HOMO-LUMO gap may also be interpreted within the

Electronic Confinement of Molecules &nor

Canlpirr

+*+ Ii +

Acid Sile

J. Phys. Chem., Vol. 98, No. 42, 1994 10867 hna

Complex

Acid SBl

-

#-\

I

*

e-

6.001

3t’

Figure 7. Qualitative orbital correlation diagram for the formation of a donor-acid site complex: on the left, when unconfined; on the right,

“

’.

t 1.oot

when confined. theoretical development of the hard-soft acids and bases principle performed by Parr and Pearson.lO By taking into account that global hardness of a molecule may be improved by the semidifference among its LUMO and HOMO energy,1° it is expected that the confined system will be softer than the free one (i.e., more reactive). This will have significant implications on the activity and selectivity of the catalyst. The fact that the gap between the n orbitals decreases (see Figure 6) implies that the excitation of the electrons will be generally easier when confining the molecule. Thus, even if no interaction occurs among the guest molecule and any active center of the catalyst, one expects that a large number of photochemical reactions ought to be favored when performing them in the cavities of molecular sieves.” On the other hand, the general increase of the energy levels of the molecules when confined will have important chemical consequences. Consider for instance the presence of acid hydroxyl groups pointing into the cavity. Their orbitals will not be affected by the confinement because they are delocalized into the lattice bulk (especially the LUMO that determines in a great part their acidic properties).12 However, a guest molecule, when residing in a cavity sees its energy levels increase, and therefore the tendency toward giving electrons to the acid site will increase too. In Figure 7 we depict the orbital level diagram for an interaction between a donor guest molecule and the catalyst acid center in the hypothetical case in which the interaction occurs out or inside of a cavity. In this case one can say that the molecule will be extrinsically more reactive when it is confined. This would partially explain the strong acidity shown by the zeolites during reactions, even though quantum chemical calculations find a relatively low ionicity and a high activation energy of deprotonation for the bridging hydroxyl groups.13 Moreover, it must be taken into account that the electronic confinement, even though different conceptually, has a similar effect on adsorbed molecules as negative electric fields inside the zeolite cavities. In both cases a destabilization of the electron energy levels is produced. Recently, a theoretical study14 has been performed which accounts for the influence of the Coulombic field produced by the zeolite framework on the acid site reactivity. The Madelung energy was explicitly included in quantum calculations by introducing the potential generated by an infinite network of charges. It has been observed there that the protonation of NH3 is favored in the situation where the electrostatic potential was not considered. Thus, the Coulombic fields together with the electronic confinement will produce an increase on the orbital energy of the guest molecule, increasing therefore its basicity and/or its reactivity. The activation of the molecules when residing in cavities of similar size has also important consequences for reactions where orbital control can play a meaningful role, for instance, the electrophilic alkylation of aromatics, where the contribution of orbital control increases when the difference in energy between the LUMO of the acceptor (alkylating agent) and the HOMO of the electron donor decreases” (see Figure 7). Since the

\

9

t

W

.

0

1.5

I

\

5.001

\

-*--2.0

2.5

3.0

3.5

4.0

Effective Cavity Diameter

4.5

5.0

5.5

(A)

Figure 8. Variation of the Coulombic and resonance integrals (Aa and AD, respective1 ) with the effective cavity size. Effective size should be 2.0-3.0 less than the crystallographic size (see the text).

1

electron confinement effect produces an increase in the energy of the HOMO of the aromatic molecule, this will involve a larger contribution of the orbital control when the reaction occurs inside the cavities of zeolites. Furthermore, the closer the size of the aromatic molecule and the cavity, the larger the increase in the energy of the HOMO, and therefore the larger the contribution of the orbital control, with the corresponding implications in paralortho selectivities.12J6 To determine the order of magnitude of the electron confinement effect in molecular size cavities, A a and AP values were approximately evaluated (for details see Appendix In). The results of such calculations are shown in Figure 8, in which A a and AB are depicted versus of the effective pore diameter when the molecule is located in the center of the cavity. It can be observed in Figure 8 that the electron confinement effect becomes important when the effective cavity size is around 3 8, or less. This cavity size does not represent the distance between two opposite atoms of the ring, as it is determined by crystallographic methods, but it should be taken as the distance among the barriers produced by the electron clouds around the nuclei. This is supported by recent calculations on purely siliceous Mordenite, performed at the periodic Hartree-Fock 1 e ~ e l . lThere, ~ the electrostatic potential map into the channel was calculated. It was found that potential energy is virtually constant, except in the vicinity of the walls, where the electrostatic field generated by the cations predominates, and especially by the electron clouds that surround the oxygens located on the border of the cavity. One can take, for instance, the “thickness” of the cavity wall as a value near 1.3-1.4 8, (which is the mean van der Waals radius of the bridging 0). This implies that one should subtract about 2.0-3.0 8, from the crystallographic size to obtain the effective one. Thus, we conclude that the electron confinement effect will be significant with cavities of 5-6 8, or less of crystallographic size. Finally, we want to stress that the electronic confinement does not only occur when the size of the guest molecule is similar to the size of the zeolite cavity. Indeed, when molecules are adsorbed in “pockets” of larger cavities, an asymmetric confinement will exist by molecules being much closer to one “wall” than to the other one. So, the motion of electrons will be more restricted in one direction (closer to the wall of the zeolite) than in the other direction. It has been observed in a previous study5s7that asymmetric localization of a system with respect to the walls leads to an increase of the energy level splitting for a given cavity size. Thus, one can expect that in this case the effect should be visible even for larger cavities. In addition, the polarization of the atomic orbitals produced by the asymmetric confinement and consequently the induced dipole moments should contribute to a larger destabilization of the

10868 J. Phys. Chem., Vol. 98, No. 42, 1994

Zicovich-Wilson et al.

molecule bond.5a This would support the importance of the surface curvature2and even the “pockets” at the external surface on molecule activation, and therefore on chemical reactiviy. In conclusion, after this study it becomes clear that an electronic confinement of the guest molecules should exist when they are inside the cavities of zeolites. This effect will be larger when the cavity and the molecule are closer in size. This produces an increase in the energy of the orbitals especially of the HOMO, besides a decrease of the intrinsic HOMO-LUMO gap of the molecule. This can be understood as producing an activation while increasing the orbital contribution in a given reaction, and at the same time decreasing the strength of the bond of the guest molecule. Moreover, the concept of electron confinement is useful to explain the stronger than expected acidity shown by zeolites. Indeed, because of the electron confinement effect, the “basicity” of the guest molecules increases and produces a stronger acid-base interaction between the bridging hydroxyl groups and the guest molecule. In addition, this effect, while being stronger on smaller cavities, may also exist in larger cavities when the molecule is adsorbed in “pockets”, indicating the importance of surface curvature on molecule activation.2fsg Since the higher the energy of the donor orbitals (especially the HOMO), the more favored electron transfer to form a covalent bond will be, owing to the orbital energy level increase in the ethylene molecule (and presumably in any electronic system), when confining it to a zeolite cavity, its basicity, understood as its ability to give an electronic pair, will increase.

Acknowledgment. We thank Profs. J. Planelles and W. Jaskolski for useful discussions on the confinement effect. CIUV are gratefully acknowledged for computational facilities. C.Z. thanks ITQ for a scholarship. Appendix I

2m

+ V(X); x E R3

(AI.2a)

= E’Y’

(AI.2b)

Realize that the solution Y’ is not necessarily quadratically integrable in all R3 but inlthe volume inside S (we call it V). Let’s now define the Q operator. “f’ being an arbitrary function,

f:R3 then

-

C,

v W.3)

if X G V

It is easy to prove that when applying Q on the confined solution Y’ one obtains a function which is quadratically integrable in all R3. Then we can write the following identity by taking account of the hermiticity of H (H is hermitian by definition, independent of the form one uses to represent it (differential, integral, or matricial))

(YlfilQY’) = (kYlQY’)= c(Y1QlY’)

(AI.4)

and it can also be written as

(YIHIQY) = (YlHQlY’) = (YIQHIY’) (Yl[k,Q]\Y’)=(Y1QIY‘)d

+

+ (Yl[kQ]lY’) (AIS)

where the square bracket indicates the operator commutator: [A,B] = AB - BA From (AI.4) and (AIS) we can obtain

The denominator in eq AI.6 is simply the integration within the volume V which is comprised inside the S surface,

(YIQIY’) = Jvdz Y*(x)Y’(x)

(AI.7)

where the star * indicates complex conjugation, whereas the numerator can be solved by using the second Green formula. First we separate the potential and kinetic energy term of H

(AI.l)

Depending on the boundary condition one imposes to the eigenfunctions we have a different eigenvalue problem for the same Hamiltonian H. Particularly, when studying free systems, one requires the eigenfunctions to be either continuous and second derivable, and quadratically integrable functions, which implies that they should vanish at the infinitum. Solutions for the same system but confined into a closed sucace S, may be obtained by solving the eigenvalue equation for H but imposing the solution being null at the surface S. So, we call Y a wave function of a given state for the free system and Y’ the one of the equivalent state for the confined system in S. Both accomplish the Schoedinger equation: *=EY

=O

if x E

The second term in eq AI.8 vanishes because, being both local operators, V and Q commute. On the other side the first term, which contains the Laplacian operator may be developed as

Let H be a one-particle Hamiltonian

tiH = - -V2

&x> = A X >

h2

= - -{Jvdr 2m

(f72Y*(x))Y’(x)Jvdz Y*(x)(~~~Y’(x))} (A1.9)

Then, by applying the second Green formula it results that

hdry*(x>(aNy’>(x>}

lo)

where integration is along surface S and & denotes directional derivative respect to vector N. Notice that he first term of eq AI. 10 must vanish because Y’(x) is null in every point of surface S (boundary condition). Finally, use of eqs A1.6, AI.7, AI.8, and AI.10, and by taking care with the terms that vanish, leads to

A€=--

h2

Adsy*(x>(aNy’)(x>

(AI. 11)

2m Jvdz Y*(x)Y’(x) Appendix I1 Let us suppose one takes the system pictured in Figure 4b. Y being a molecular orbital for the unconfined x system and

Electronic Confinement of Molecules

J. Phys. Chem., Vol. 98, No. 42, 1994 10869

Y’ the equivalent one for the confined system, one can obtain the electronic confinement energy A6 by applying eq 2 in the text: A€ =

2

[-Jm --dw dy ~,Y?(x,y,d)Y(x,y,d) +

where N is a normalization factor, Z is the effective atomic charge of the carbon atom, ao is the Bohr radius, and r, and rb represent the distances of the electron with respect to nuclei a and b. So, an adequate weight function which allows an available improvement of n’(x,y,z) (see Figure 4b), might be (AIII.3)

where a, indicates partial derivative with respect to z direction and the energy is expressed in atomic units. By taking account of the system symmetry, eq AII.l leads to

where d has the same meaning as in Figure 4b. By using eq AIII.1, the unnormalized 2p, atomic orbital for the confined system will be exp(-Zr:r$ao)

n

By using eqs 1 and 2 in the text, one can express eq AII.2 as a function of the pz atomic orbital of the carbon atoms: a(x,y,z) and b(x,y,z) for the unconfined molecule and a’(x,y,z) and b’(x,y,z) for the confined one. Then, the energy splitting of the two molecular orbital states JC and JC* is expressed as

(AIII.4)

Such function accomplishes the conditions stated above, that is, condition 1, a’(x,y,fd) = 0; condition 2, lim- a’(z,y,z) = a(z,y,z); condition 3, limz+’(z,y,z) = (dh)sin(nz/d)),which is the corresponding solution of the free particle in a onedimensional box of size 2d. Then we can obtain approximately A a and Ap by using eqs 3c, 3d, and AIII.4,

. - .

Aa =

(AIII.5a)

aiaj

cvi+ Pj)’

‘ij

ij

and which is equivalent to eq 3 in the text. To obtain eq AIL3 we have to suppose null differential overlap of the orbitals, that is:

hd&J-:dw

dy a’(x,y,z)b(x,y,z) e 0

d

q

z cxp[

-vi+ Pj)d2

-

”1

Pi

AB =

+Pj

(AII.4)

according with the HMO theory.6 where ai and pi come from the expanding the exponential part of the Slater function in terms of Gaussians:

Appendix I11 The evaluation of the energy increment by means of eq 2 is sometimes difficult because the confined wave function Y’(x) is generally unknown. However, one can improve it by a new function which is obtained as any weight function times the exact unconfined solution:

where w(x) is the weight function and Y(x) is the wave function corresponding to the equivalent state as Y’(x)on the unconfined system. To give an effective improvement, one should impose Y ( x ) to accomplish some restrictive conditions: (1) Y ( x ) must vanish at the cavity border. (2) When increasing the size of the cavity, Y’(x) must tend asymptotically to Y(x). (3) When the potential function of the system Hamiltonian approaches zero, Y ( x ) should tend to the corresponding solution of the free particle in the cavity. In our problem Y(x) is a JC function which may be written as a linear combination of two 2pz Slater-type atomic orbitals, that is:

+

n(x,y,z) = a(x,y,z) b(x,y,z) = Nz[exp(-Zra/uO)

+ exp( -Zr&,)]

(AIII.2)

exp(-Zr,/ao) =

Eaiexp[-Pir:]

(AIII.6)

i

In eqs AIII.5a and AIIISb, “b” represent the distance between atoms a and b, and Zij denotes the following integral:

Zu =

hd:z

+ Pj)z23 dz = hdc o s E ) exp[-vi + Pj)z2] dz (AIII.7)

s i n E ) exp[-vi

Equation AIII.5 allows us to evaluate approximately the boxing effect on the ethylene molecule JC system for different cavity sizes. Results of such calculations are depicted in Figure 8. Equation AIII.6 was improved by using the standard STO3G expansion’* and integrals in eq AIII.7 were evaluated numerically.

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