Quantum Confinement of the Covalent Bond beyond the Born

Article pubs.acs.org/JPCB

Quantum Confinement of the Covalent Bond beyond the Born− Oppenheimer Approximation A. Sarsa,*,† J. M. Alcaraz-Pelegrina,† C. Le Sech,‡ and S. A. Cruz¶ †

Departamento de Fı ́sica, Facultad de Ciencias, Universidad de Córdoba, E-14071 Córdoba, Spain Université Paris Sud 11, CNRS, Institut des Sciences Moleculaires d’Orsay-ISMO (UMR 8214), 91405 Orsay Cedex, France ¶ Departamento de Fı ́sica, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534, 09340 México, DF, México ‡

ABSTRACT: Dirichlet boundary conditions with different symmetries, spherical and cylindrical impenetrable surfaces, are imposed on the covalent electron pair of a molecular bond. Accurate results for different observable like energy and interparticle distances are calculated using quantum Monte Carlo methods beyond the Born−Oppenheimer approximation. The spherical confinement induces a raise in the bond energy and shortens the internuclear distances even for a relatively soft confinement. When cylindrical symmetry is considered, similar qualitative behavior is observed though only the electrons are confined. A compression followed by a relaxation process of the confined bond is shown to induce a vibrationally excited state. Finally, a brief qualitative discussion based on a simplified picture of the role of compression/relaxation cycles in enzyme catalysis is given.

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INTRODUCTION The effect of spatial confinement on the physical and chemical properties of atoms and molecules has been a subject of increasing research interest in recent years. These properties are known to be quite different for an atom or molecule submitted to spatial constraint as compared with their free counterparts. It is therefore important to develop models of quantum confinement to analyze the effects induced by a constraint on a scale comparable to the atomic size. The problem of quantum confinement is far from being of just academic interest. Indeed, during the past 70 years, this problem has been intensively addressed in a number of fields in physics and chemistry as well as in other areas such as astrophysics. Within the class of physical problems treated from this perspective, we mention, for instance, the effect of pressure on the energy levels, the polarizability and ionization threshold of atoms and molecules, and the electronic and optical properties of artificial atoms like quantum dots and wires in nanostructured semiconductor materials. We refer the interested reader to available recent reviews.1−4 On the chemistry side, quantum confinement effects have also attracted much attention with increasing active research in the study of catalytic reactions along zeolitic nanochannels and molecular insertion in fullerene-like structures by efficient new techniques; see, for example, ref 5 and references therein. At this stage, it is worth pointing out that the catalytic chemical transformations made by enzymes, so important for living systems, all have in common a feature that is the confinement of substrates inside of the enzymes’ cavities to give subsequently the products. Hence, supramolecular cavities have been proposed to mimic the catalytic properties of enzymes.6 Spatial confinement effects on atomic and molecular properties have been modeled through use of appropriate confining barrier potentials dependent on the physical © XXXX American Chemical Society

conditions of the host medium where the atom or molecule is embedded. The class of confining model potentials used ranges from harmonic-type up to step-like. The former class has been used to mimic a smoothly varying confining strength, for example, electrons or hydrogenic impurities in quantum dots,7−9 while the latter type corresponds to a more localized strength due to the polarization or presence of repulsive charges within the confining boundary, for example, a fullerenelike cage. Confinement models based on the step-like barrier potential have been widely used to mimic the behavior of atoms or molecules spatially limited by either penetrable or impenetrable open or closed boundaries. This kind of model allows also to study the effect of piecewise-defined confining potential barriers, for example, adding an attractive layer potential of finite depth at the boundary, as done recently in the study of giant photoionization resonances of endohedrally caged atoms, ref 10 and references therein. In the case of closed boundaries, these are the so-called box models. In the present work, the box model for impenetrable spherical and cylindrical symmetries will be considered. Hence, confinement is viewed as a restriction to the space available for the particles alone. The corresponding quantum mechanics problem is to solve the time-independent Schrödinger equation with Dirichlet boundary conditions. Of course, in real situations, the wall is never fully impenetrable. However, this hard-wall approximation has been widely used because it provides a first useful means to explore the physically relevant changes in the system properties due to confinement. It is Received: March 19, 2013 Revised: May 22, 2013

A

dx.doi.org/10.1021/jp402727b | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

Article

Mi i = 1, 2 M1 + M 2 + m1 + m2 mi i = 1, 2 βi = M1 + M 2 + m1 + m2

similar to the usefulness of the blackbody analysis, though the perfect blackbody does not exist. Here, we are specifically concerned with the change in the chemical bond when constraints are imposed on the covalent electron pair. The theory of the chemical bond interprets the link between two atoms as the result of the fine balance between electrostatic interactions and exchange forces. In most of the theoretical approaches describing molecular systems by quantum chemistry, the electron pair of the covalent bond is allowed to span the whole space without any constraint. However, it is well-known that the probability density associated to the wave function is larger between the nuclei in the ground Σg state. This raises a pending question: what would be the modifications induced in the molecule when the electrons are subject to external constraints that change the available space where they can be located? An obvious direct consequence will be the modification of the probability distribution of the bonding electrons. The hydrogen molecule is one of the best examples of molecular covalent binding. It is thus an attractive system to study changes in the properties of the covalent bond when the electrons and nuclei are confined by an external constraint. The study of this two-electron molecule is relevant for the understanding of larger molecular systems containing multiple covalent bonds, including their dynamics. For this reason, a lot of papers have been devoted to study the confined hydrogen molecule.11−14 Noteworthy, all of these studies have been carried out within the framework of the Born−Oppenheimer (BO) approximation; hence, the effect of confinement on the combined electron−nuclei dynamics for this system is still an unresolved problem. The present paper analyzes, beyond the BO approximation, the change in the covalent chemical bond when the simultaneous confinement of electrons and nuclei is considered. Hard boundary conditions, spherical box and cylindrical axial symmetry, are chosen in this paper to simulate the external constraint. Finally, a discussion is presented on the possible role of bond compression/decompression cycles in the active site of enzymes as a mechanism to the enzymatic catalytic activity.

αi =

fi =

T=− −

(3)

ℏ2 2 ℏ2 2 ℏ2 2 ℏ2 2 ∇R c − ∇R − ∇r1 − ∇r 2M 2μR 2ε1 2ε2 2

ℏ2 ∇r ·∇r M1 + M 2 1 2

(4)

where each of the differential operators is evaluated with respect to the Jacobi coordinates employed in this work. The last term in eq 4 corresponds to the mass polarization contribution. The following definitions hold M = M1 + M 2 + m1 + m2 μR = εi =

M1M 2 M1 + M 2 mi(M1 + M 2) mi + M1 + M 2

i = 1, 2

(5)

In our case, our four-body problem is the hydrogen molecule, hence M1 = M 2 ≡ Mn

m1 = m2 ≡ m

ε1 = ε2 ≡ ε

(6)

and m ≪ Mn

so that the contribution of the mass polarization term to the ground-state energy is very small, and it will be considered here perturbatively. The following mass values have been employed Mn = 938.2720290 MeV/c 2 m = 0.510998918 MeV/c 2

(7)

This gives the following values for μR and ε in atomic units of mass (m = 1)

THEORY Let us consider the four-body problem formed by the two nuclei together with the two electrons. The kinetic energy operator, T, for this four-body problem is

μR = 918.076336 ε = 0.99972777

(8)

The potential energy operator, V, for a confined H2 molecule is

(1)

where Mi and Ri are the masses and position vectors of the nuclei and mi and si stand for mass and the position vectors of the electrons with respect to an arbitrary origin. ∇2Ri and ∇2si are the Laplacians with respect to the coordinates of the nuclei and the electrons, respectively. The following Jacobi coordinates are used here

V=

1 1 1 1 1 1 − − − − + + Vconf R r1A r1B r2A r2B r12

(9)

with ri A = |ri − R1|

ri B = |ri − R 2|

i = 1, 2

(10)

and R = |R|

R c = α1R1 + α2 R 2 + β1s1 + β2 s 2

r12 = |r1 − r2|

(11)

Vconf stands for the confinement potential giving the Dirichlet condition at the boundary limits. With the Jacobi coordinate system here employed, the center of mass motion can be separated. Then, the final form of the Hamiltonian for the H2 molecule beyond the BO approximation here considered is

R = R1 − R 2 r1 = − f1 R1 − f2 R 2 + s1 r2 = − f1 R1 − f2 R 2 + s 2

i = 1, 2

After the change of coordinates, we obtain

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ℏ2 2 ℏ2 2 ℏ2 2 ℏ2 2 T=− ∇R1 − ∇R 2 − ∇s1 − ∇s 2M1 2M 2 2m1 2m2 2

Mi M1 + M 2

(2)

with B


The Journal of Physical Chemistry B H=−

ℏ2 2 ℏ2 2 ℏ2 2 ∇R − ∇r1 − ∇r + V 2μR 2ε 2ε 2

Article

cylinder of radius ρ0. The axis of the cylinder is taken to the z axis. The cutoff function becomes

(12)

⎛ ⎛ρ + ρ ⎞ ρ ⎞⎛ ρ⎞ 2 ⎟⎟ W(ρ1 , ρ2 ) = ⎜⎜1 − 1 ⎟⎟⎜⎜1 − 2 ⎟⎟ exp⎜⎜ 1 ρ ρ ρ ⎠ ⎝ ⎝ 0 ⎠⎝ 0⎠ 0

with V given in eq 9. The Schrödinger equation to be solved is H Ψ(r1, r2, R) = E Ψ(r1, r2, R)

(13)

with

with H given in eq 12. The wave function Ψ employed here to describe the ground state of a confined H2 molecule in a nonadiabatic scheme is taken as a product of several factors to take into account different physical considerations.

ρi =

(14)

EΨ =

ϕ(ri, R) = exp[−(α + βR )λi] cosh{[α − (1 − β)R ]μi }

λi =

(17)

with riA and riB as the distances of electron i to the nuclei A and B given in eq 10. The second factor in the wave function, eq 14, is a Jastrow term describing the interelectronic correlation. In this work, the following form has been employed

(18)

The factors F(R) and YK,MK(θ,ϕ) describe, respectively, the nuclear vibration and rotation of the molecule. The lowest rotational and vibrational level will be considered here, so that K = 0 and YK,MK(θ,ϕ) = (4π)−1/2. The model function for the vibration of the nuclei is taken as a harmonic oscillator wave function, with Req as the equilibrium internuclear distance F (R ) =

exp[−δ(R − R eq)2 ] R

(19)

The last factor in eq 14 is a cutoff factor included in order to fulfill the Dirichlet conditions. Boundary hard spherical surfaces with radius R0 centered midpoint to the nuclei are considered here. The cutoff function, W, is chosen to be16−18 ⎛ r ⎞⎛ r ⎞⎛ R⎞ W (r1 , r2 , R ) = ⎜1 − 1 ⎟⎜1 − 2 ⎟⎜1 − ⎟ R 0 ⎠⎝ R 0 ⎠⎝ R0 ⎠ ⎝ ⎛ r + r2 + R ⎞ exp⎜ 1 ⎟ R0 ⎝ ⎠

⟨Ψ|H |Ψ⟩ ⟨Ψ|Ψ⟩

(23)

where H is the Hamiltonian, eq 12, and Ψ the variational wave function, eq 14. The calculation of this expectation value is carried out here by using the variational Monte Carlo (VMC) method. This method is based in the use of random walks in the configuration space of the system to calculate expectation values; see, for example, ref 19 for a complete review. The functional developed in ref 17 for the study of confined systems in VMC calculations is employed in this work to calculate the expectation value of the Hamiltonian, eq 23. In order to improve the energies of the confined H2 molecule calculated variationally, a diffusion Monte Carlo (DMC) calculation is carried out. We recall briefly here the main ideas underlying this method. Further details can be found in, for example, ref 19. The DMC method starts from the timedependent Schrödinger equation in imaginary time, which becomes a classical diffusion equation. To determine the random walk that simulates the diffusion, the Green’s function at short time approximation is invoked. Then, a step of the random walk consists of an isotropic Gaussian diffusion and branching processes of the walkers. After a large number of iterations, the excited-state contributions are projected out from the initial ensemble, converging to the ground state, and the ground-state energy can be deduced. In practical implementations, an approximate wave function is employed to bias the random walk, reducing in this way the numerical error. Very involved parametrizations, which are generally time-consuming, will slow down the calculation due to the fact that in each step, the gradient and the Laplacian must be calculated for each walker. Hence, compact, concise, and still accurate wave functions are ideal. The optimized wave functions obtained variationally are employed here as guiding functions. For the systems studied in this work, with a spatially symmetric groundstate wave function vanishing only at the boundaries, the DMC provides the exact energy within the numerical error.

(16)

where λi and μi are the confocal elliptic coordinates

⎛ br12 ⎞ J(r12) = exp⎜ ⎟ ⎝ 1 + cr12 ⎠

(22)

where Zeq is the equilibrium internuclear distance. In eqa 16, 18, and 19 for spherical confinement or eq 22 for cylindric confinement, the parameters α, β, b, c, δ, Req, or Zeq are fixed variationally. The expectation value of the Hamiltonian is

(15)

The function ϕ(ri,R) is a three-body wave function for the two nuclei and the electron i. The following choice of a simple analytical function was successful for a precedent study of the H+2 ion.15

ri A + ri B 1 ≤ λi ≤ ∞ R r − ri B μi = i A −1 ≤ μi ≤ 1 R

i = 1, 2

F(Z) = exp[−δ(Z − Zeq)2 ]

The first factor in eq 14 is Φ(r1, r2, R) = ϕ(r1, R)ϕ(r2, R)

xi2 + yi2

and consistently, the function describing the vibration in the internuclear distance, Z, is taken as a one-dimensional harmonic oscillator

Ψ(r1, r2, R) = Φ(r1, r2, R)J(r12)F(R )YK , MK(θ , ϕ)W (r1 , r2 , R )

(21)

■

(20)

RESULTS The VMC method is used first to make the variational optimization of the parameters α, β, δ, b, c, Req, or Zeq . The optimized wave function is next employed as a guiding function

When cylindrical boundary conditions are considered, the spherical symmetry is reduced to an axial one. The molecule is assumed to lie along the axis of symmetry of a nonpenetrable C



Article

Table 1. VMC, EVMC, and DMC, EDMC, Ground-State Energies for the Spherically Confined H2 Molecule for Different Confinement Radiia R0

EVMC

EDMC

Reqb

Eb

α

β

b

c

δ

Req

2.0 2.5 3.0 3.5 4.0 4.5 5.0 20 ∞

−0.47909(4) −0.8709(4) −1.02532(4) −1.09516(3) −1.12522(3) −1.13945(3) −1.14572(3) −1.15286(3) −1.15304(3)

−0.49408(8) −0.88488(8) −1.04381(8) −1.11216(9) −1.14193(9) −1.15465(9) −1.16025(9) −1.16400(4) −1.16403(5)

0.90 1.05 1.17 1.25 1.30 1.35 1.38 1.40 1.40

−0.4395 −0.8367 −1.0005 −1.0723 −1.1042 −1.1188 −1.1257 −1.1330 −1.1332

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.11 0.11

0.59 0.56 0.57 0.56 0.56 0.56 0.55 0.60 0.60

0.60 0.56 0.70 0.57 0.62 0.57 0.60 0.72 0.72

0.41 0.34 0.36 0.33 0.35 0.33 0.40 0.45 0.45

19.5 16.5 15.0 11.5 11.0 9.9 9.2 8.8 8.8

1.02 1.13 1.26 1.35 1.40 1.43 1.45 1.46 1.46

The optimum parameters in the VMC calculation α, β, b, c, δ, and Req are also shown. In parentheses, we show the statistical error in the last figure. Also reported, are corresponding variational BO calculations using the method of ref 14 (see the text). bVariational BO calculations of this work using the method of ref 14.

a

Table 2. Expectation Values for Different Characteristic Distances and Angles Considered in This Work (see text) for the Spherically Confined H2 Moleculea

a

R0

⟨R⟩

⟨ra⟩

⟨r⟩

⟨ree⟩

⟨θee⟩

⟨θ(r,R)⟩

ℏ2/Mn⟨∇1·∇2⟩

2 2.5 3 3.5 4 4.5 5 20 ∞

0.96415(1) 1.11746(1) 1.24811(1) 1.33744(1) 1.38921(2) 1.41981(2) 1.44143(2) 1.45266(2) 1.45304(2)

0.97004(3) 1.13664(4) 1.27223(4) 1.36702(5) 1.44017(5) 1.48641(5) 1.52588(5) 1.55399(6) 1.55848(5)

0.86373(3) 1.01511(4) 1.13694(4) 1.22210(5) 1.29164(5) 1.33561(6) 1.37440(6) 1.39989(7) 1.40469(7)

1.28400(5) 1.52648(7) 1.73379(8) 1.85372(8) 1.9680(1) 2.0339(1) 2.0845(1) 2.1393(1) 2.1470(1)

96.364(3) 97.577(4) 99.236(4) 98.219(4) 98.609(4) 98.320(4) 97.634(4) 98.094(4) 98.094(4)

89.999(4) 89.999(3) 90.001(4) 90.000(4) 89.999(4) 90.000(4) 89.998(4) 90.001(4) 90.000(4)

5.414(5) × 10−5 4.786(4) × 10−5 4.506(3) × 10−5 3.583(3) × 10−5 3.335(2) × 10−5 3.078(2) × 10−5 2.639(2)× 10−5 2.724(2) × 10−5 2.707(2) × 10−5

The last column corresponds to the average value of the mass polarization term. In parentheses, we show the statistical error in the last figure.

Table 3. VMC, EVMC, and DMC, EDMC, Ground-State Energies for the Cylindrically Confined H2 Molecule for Different Confinement Radii ρ0a ρ0

EVMC

EDMC

⟨Z⟩

⟨ree⟩

α

β

b

c

δ

Zeq

2.0 2.5 3.0 4.0 5.0 20

−0.79687(4) −1.01248(4) −1.09329(4) −1.14140(4) −1.15100(4) −1.15304(4)

−0.8125(5) −1.0270(1) −1.1094(3) −1.1552(4) −1.1626(3) −1.1640(1)

1.19702(2) 1.29378(2) 1.35254(2) 1.41131(2) 1.45094(2) 1.48180(2)

1.5777(1) 1.8162(2) 1.9065(1) 2.0608(1) 2.1083(1) 2.1423(1)

0.115 0.085 0.085 0.082 0.088 0.106

0.58 0.59 0.60 0.60 0.60 0.60

0.45 0.58 0.58 0.58 0.52 0.48

0.35 0.32 0.32 0.30 0.28 0.30

11.5 11.5 11.5 10.0 9.5 9.0

1.20 1.30 1.36 1.42 1.46 1.49

a ⟨Z⟩ and and ⟨ree⟩ correspond to the mean internuclear and electron−electron distances, respectively. The optimum parameters in the VMC calculation are also shown. In parentheses, we indicate the statistical error in the last figure.

both cases, it is of interest to notice the energy increase and reduction in the internuclear separation as the confinement strength increases. Even for moderate confinement, the change in energy is significant. For instance, for R0 = 4 au, the calculated DMC energy is E = −1.1419 au to be compared to the free-molecule ground-state energy EGS = −1.1640 au. This significant relative change in energy has also been noticed for the softer spherical harmonic confinement of H2.12 In Table 2, some expectation values of interest for the different confinement radii considered are displayed. ⟨R⟩ is the expectation value of the internuclear distance, ⟨ra⟩ is the average electron−nucleus distance, ⟨r⟩ is the expectation value of the electron distance to the nuclear center of mass, and ⟨ree⟩ is the average value of the interelectronic distance. Finally, ⟨θee⟩ and ⟨θ(r,R)⟩ are, respectively, the expectation value of the angle between r1 and r2 and the expectation value of the angle subtended between the position vector of each electron and the molecular axis. For completeness, we have included in this table

in the DMC approach. A check of the accuracy can be made easily considering the calculated ground-state dissociation energy D0 of the free H2 molecule, D0 = 36120 ± 11 cm−1 to be compared to the experimental value20 D0 = 36118 cm−1; see, for example, ref 21 for a very accurate calculation including nonadiabatic, relativistic, and quantum electrodynamic corrections. In the case of spherical confinement, Table 1 shows the VMC (EVMC) and DMC (EDMC) energies for the ground-state energy values for different confinement radii R0. We also include in this table the corresponding equilibrium internuclear distances and the values of the variational parameters employed in each case. For comparison purposes, we also show the variational BO energies and equilibrium internuclear distances calculated using the method reported in ref 14 for the H2 molecule confined by a quasi-spherical hard prolate ellipsoid with interfocal distance D = 0.1 au. While the latter calculations do not include electronic correlations, a close correspondence with the accurate VMC and DMC calculations is observed. In D



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selected set of confining radii for both spherical (inverted triangles) and cylindrical (squares) boundaries. All values are taken from Tables 1−3. For the sake of comparison, the BO curve representing the free H2 molecule ground-state energy versus fixed internuclear distances R, with equilibrium bond length Req = 1.46 au, is also shown. Clearly, a steep raise in energy is obtained when either spherical or cylindrical confinement is set. Moreover, the constraint allows for a metastable state with higher energy of the molecule than in its free state. It is of interest to analyze the fate of the metastable state when the constraint is removed. For instance, let us consider the metastable state represented by label M in Figure 1. Let us further assume a sudden relaxation of the constraint. The symmetry of the metastable state (g) is identical to that of the free state. This precludes an electronic radiative dipolar transition. Thus, assuming that the removal of the constraint is ideally sudden, the final state will be a vibrationally excited state of the 1Σ ground state of the molecular bond represented by the horizontal broken line in Figure 1. The question of how such metastable state can be obtained in practice with the help of a suitable confinement is of importance. To achieve a significant decrease in the average bond length following the electronic confinement, it is necessary that the forces are set on the bonding electrons during a sufficiently long time in order for the nuclei, which are heavy particles, to have enough time to move. Such a long constraint in time cannot be obtained in, for example, a strongly polar solvent because the forces due to the electrostatic environment are not coherently reoriented during a sufficiently long time due to the thermal dispersion in continuum solvent models. Confinement is probably not a relevant mechanism of the chemical reactions in water solutions. However, in polar cavities, where a stable preorganized electrostatic environment may be found, the right confinement conditions on the molecules may be set to favor chemical reactions. This idea is in line with one of the common features in the catalysis by enzymatic transformations, namely, the confinement of the substrate in the enzyme pocket. Indeed, compression of the nuclei of the substrate inside of the cavity is often invoked to contribute to the catalytic power of the enzymes by lowering the distance between reactive termini, thus facilitating nuclear tunneling across the activation barrier.23,24 Different works concerned with the problem of catalytic transfer of hydrogen atoms in red−ox reactions argue that the distance between the donor and acceptor is decreased by compression.25 In this connection, the results of this work may be useful to provide heuristic arguments on the role of confinement in this highly complex process as follows. We suggest that a compression/relaxation cycle by electrostatic forces might occur inside of the polarized active site of the enzyme, creating a microenvironment leading to an enhancement of the constraint applied to the electrons of a chemical bond. Thus, the net result is a vibrationally excited molecular bond,26 equivalent to an increase in temperature of the substrate, facilitating the subsequent atomic rearrangement to give the products. Because our treatment involves also confinement effects on the nuclei, we might expect isotopic differences to appear in the vibrational transition states for, for exampe, H atom transfer from a C−H bond as compared with the C−D case,27,28 thus leading to different reaction rates. This mechanism would help understanding the role of confinement

the nominal value of the mass polarization term (see eq 4) for the various confinement radii. As expected, hard-wall confinement of the bonding electrons is able to shorten significantly the internuclear distances. However, here, we give numerical reference of such an effect within our non-BO approach. For instance, for R0 = 2 au, the average internuclear distance is ⟨R⟩ = 0.964 au to be compared with ⟨R⟩ = 1.453 au for the free molecule. This result was already noticed by Segal et al.22 in relation with the cold fusion problem. On the other hand, the values of the angles ⟨θ(r,R)⟩ remain in all cases close to 90°, illustrating that the bonding electrons stay on the average in the midplane perpendicular to the molecular axis. Note that the contribution of the mass polarization term is indeed very small in all cases, although with a steadily increasing behavior as the confinement radius is reduced. Table 3 displays the VMC and DMC ground-state energy values of the H2 molecule confined by a hard cylindrical boundary of varying radius. We note here that in this case, only the electrons are submitted to radial confinement, as indicated by eq 21. Note, however, that the cylindrical hard boundary constraint induces a significant raise in energy, even for a relatively moderate confinement condition, as already noted in the case of the spherical confinement. Also, these results show a decrease in the mean internuclear distance as the cylindrical radius is reduced. This behavior occurs even if the Dirichlet boundary conditions are only imposed along the radial direction while no restrictions are imposed along the axial one. Note also that the cylindrical constraint needs a stronger condition, that is, ρ0 < R0, in order to reach the same energy increment as that due to spherical confinement. Figure 1 shows the DMC equilibrium energy values plotted against the average internuclear distances ⟨R⟩ or ⟨Z⟩ for a

Figure 1. DMC equilibrium ground-state energy values for the H2 molecule confined by spherical and cylindrical hard boundaries with specific confinement radii R0 (inverted triangles) and ρ0 (squares), respectively. The corresponding average equilibrium internuclear distances are given by the horizontal axis. The internuclear distance dependence of the ground-state BO energy for the free H2 molecule is also shown for comparison. E



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on some primary and secondary kinetic isotopic effects (KIEs) observed in catalyzed hydrogen-transfer reactions.29,30 We emphasize here that the above analysis is still a bit speculative concerning the complex mechanisms inherent in the catalytic activity of enzymes. However, the results of our study of the confined H2 molecule may serve as a clue to consider the role of confinement/relaxation cycles for the induction of vibrational excitation of a molecular bond specific to a chemical reaction of the substrate, which could be an important electronic mechanism in the catalytic power of enzymes.

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CONCLUSION The ground-state energy and equilibrium bond length behavior of the H2 molecule confined by hard spherical and open-ended cylindrical boundaries have been studied beyond the BO approximation using the variational and diffusion Monte Carlo methods. The non-BO results reported in this work constitute a benchmark reference calculation, even if the BO calculations may provide similar results. Our calculation considers the confinement effect on the coupled motion of nuclei and electrons and serves as a reference on the validity of the BO approximation for the treatment of the confined system. It is found that in all cases, confinement by hard surfaces of the bonding electrons augments the energy and decreases the interparticle distances. This holds even if the impenetrable wall is not necessarily a closed surface, as illustrated by the results obtained for the cylindrical symmetry. Due to confinement, a metastable ground state with higher energy as compared with the free-molecule is obtained. This raises the interesting question of the fate of such a metastable state when the constraint is suddenly removed. We have shown that a molecular vibrational transition from the confined metastable ground state to its free counterpart is induced when a confinement/deconfinement cycle is considered. This mechanism has been considered as a simplified picture to qualitatively visualize the role of the compression/relaxation cycles in the complex catalytic behavior of enzymes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.S. and J.M.A.-P. acknowledge partial financial support by the Spanish Dirección General de Investigación Cientı ́fica y Técnica (DGICYT) under Contract FIS2012-39617-C02-02 and by the Junta de Andalucı ́a.

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REFERENCES

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Pre-Equilibria. Philos. Trans. R. Soc. London, Ser. B 2006, 361, 1399− 1415. (28) Pollak, E. Quantum Variational Transition State Theory for Hydrogen Tunneling in Enzyme Catalysis. J. Phys. Chem. B 2012, 116, 12966−12971. (29) García-Viloca, M.; Truhlar, D.; Gao, J. Reaction-Path Energetics and Kinetics of the Hydride Transfer Reaction Catalyzed by Dihydrofolate Reductase. Biochemistry 2003, 42, 13558−13575. (30) Northrop, D. Unusual Origins of Isotope Effects in EnzymeCatalysed Reactions. Philos. Trans. R. Soc. London, Ser. B 2006, 361, 1341−1349.

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Quantum Confinement of the Covalent Bond beyond the Born

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