MOLECULAR STRUCTURE. The general molecular Schrödinger equation, apart from electron spin effects, is. nn ee en

Transcription

1 MOLECULAR STRUCTURE The Born-Oppenheimer Approximation The general molecular Schrödinger equation, apart from electron spin effects, is ( ) + + V + V + V =E nn ee en T T ψ ψ n e where the operators in the Hamiltonian are the kinetic energy operators of the nuclei and the electrons, and then the potential energy operators between the nuclei, between the electrons, and between the nuclei and electrons.

2 Coordinates are defined below for a three nucleus, three electron molecule. Definition of distances in a molecule

3 In general, the explicit forms of these operators are as follows nuclei = n M α α α T T e electrons = m i=1 e i V nn nuclei α β = α β> α Z Z e R αβ V ee e = i=1 j>i r electrons ij V en nuclei electrons = α i There is a repulsive interaction among the nuclear charges and a repulsive charge-charge interaction among the electrons. However, the interaction potential between electrons and nuclei is attractive, since the particles have opposite charges. This particular interaction couples the motions of the electrons and the motions of the nuclei. Z e α r i α

4 T n = α α M α nuclei T e = electrons m i=1 e i V nn = nuclei α β> α ZZe α β Rαβ V ee = electrons e r i=1 j>i ij V en = nuclei electrons α i Zα e riα

5 Born-Oppenheimer Approximation The wavefunctions that satisfy this Hamiltonian must be functions of both the electron position coordinates and the nuclear position coordinates, and this differential equation is not separable. In principle, true solutions could be found, but this is a formidable differential equation to work with. An alternative is an approximate (but generally a very good approximation) separation of the differential equation based upon the very large difference between the mass of an electron and the masses of the nuclei. The difference suggests that the nuclei will be sluggish in their motions relative to the electron motions. Over a brief period of time, the electrons will "see" " the nuclei as if fixed in space; the nuclear motions will be relatively slight. The nuclei, on the other hand,,will "see" the electrons as something of a blur, given their fast motions. We can thus solve for electronic properties with fixed nuclei.

6 ( T + T + V + V + V ) Ψ( R, r, ) =E Ψ( R, r, ) nn ee en 1 1 n e So when the Born-Oppenheimer approximation is invoked, Ψ( Rr,,...) = c φψ φ Rψ r,... ( ) R ( ) 1 ij i j i j 1 i j Electronic: [T e + V ee + V (R) en] ψ (R) (r,r,...) = E (R) (R) 1 ψ (r 1,r,...) Nuclear: [T n +(V nn + E (R) ) E] φ = 0 R represents nuclear coordinates r i represents electron coordinates

7 It is the Born-Oppenheimer approximation that leads to the concept of potential ti energy surfaces!

8 Potential Energy Surfaces A potential energy surface is a representation of a function of all the internal coordinates of a molecular system. For even a simple molecule such as ammonia, the potential ti is a function of six coordinates, making is difficult to display or to visualize. We may, however, make a graphical representation of a slice through the multidimensional surface. The slice is the potential function with all but one or two of the coordinates fixed at certain chosen values. If all but one are fixed, the slice is a function of only that one coordinate and its representation is a potential curve. If all but two are fixed, the slice is a function of two coordinates. As an example, look at linear HCN Contour Plot of HCN Ground State

9 Functions of two coordinates are easily represented by contour diagrams. This is a very common method to present potential energy surface slices, as shown in the previous picture. The curves that we see in this figure connect equipotential points on the potential energy surface. Cutting across one of these curves means going "uphill" or "downhill" in energy. Usually, we may identify a lowest energy point on a slice though a potential energy surface (i.e., a two-dimensional contour plot), and as above, that point lies near the middle of the diagram. The contour plot can only show us that the potential is uphill in two directions away from this point. In the other directions, the ones that were fixed to create the slice, the surface might be sloping upward or downward. Should it be that the surface slopes upward in every direction, i then the point is a minimum energy point or minimum on the surface. For complicated surfaces, s it is possible for there to be several minima, and the one of lowest energy is called the global minimum. The global minimum is a point on the surface that corresponds to the equilibrium structure of the species.

10 A slice of a hypothetical potential energy surface of a hypothetical linear triatomic molecule ABC is shown below. One can envision several possible chemical processes taking place in this collinear configuration. This diagram extends to the regions where the molecule is dissociated into the diatomic AB and atom C and where it is dissociated into the diatomic BC and atom A. The minimum is near the lower left. Stretching either bond, which means increasing either coordinate, is an uphill process. However, there is a unique point in the A-B stretching where the potential will start to be downhill. This point, called a saddle point, is where the potential surface is uphill in each direction except one, and in that one direction, the potential is downhill, either forward or backward. Fig. Contour Plot of a Slice of a Representative Molecule ABC.

11 If we follow the contour plot of Fig to the right of the saddle point, we encounter a trough. This is simply a region where the potential is slowly changing, either upward or downward, along one direction. Eventually, this trough will have a flat bottom and walls that do not change with the A-B separation as the point is reached where the A atom is completely removed from the BC diatomic and there is no interaction. At this limit (far right of Fig. ), a cut through this two-dimensional surface for any A-B distance is a potential curve; it is the potential energy as a function of the B-C separation, which is just the stretching potential of the BC diatomic. Likewise, a horizontal cut across the top left of the surface in Fig. must be the stretching potential for the AB diatomic molecule. l

12 The bottom of a trough that connects limiting regions such as these with minima or with saddle points is called a minimum energy path. That is a good term, especially if we consider the potential surface as if it were a physical surface with real hills and valleys. If we were at the bottom of the trough on the far right of the surface in Fig. and wished to "hike" to the equilibrium structure on the lower left, the minimum climb would take us right through h the saddle point and then down the valley where the potential minimum is found. That is the minimum energy path. Notice that it does not follow any one coordinate direction. At the outset, it follows the direction of the coordinate of the horizontal axis. Then it twists a bit, and if we follow it further, it will follow the direction of the coordinate of the vertical axis.

13 Linear ABC and the Reaction Path Concept The bottom of a trough that connects limiting regions such as these with minima or with saddle points is called the minimum energy path.

14 Energetic profiles for reactions, interconversions, and isomerizations are often given as simple curves of potential energy (vertical axis) versus a reaction coordinate (horizontal axis). This is not as abstract a notion as it may seem, because the minimum energy path provides a perfectly acceptable reaction coordinate. That is, each step along a reaction coordinate is a step along the minimum i energy path. In this sense, energy profiles are simply special slices through potential energy surfaces. Again, the Born- Oppenheimer approximation has provided the basis for an important chemical concept, and it provides the context for discussing details of reaction energetics.

15 The H + ION We have seen how the Born-Oppenheimer Approximation allows us to separate nuclear motion from electronic motion and leads to the concept of molecular potential energy surfaces. This concept is at the heart of all of our understanding of chemical processes. With this approximation made, we then attempt to solve the one-electron molecule, H +. As with the H atom, these wave functions will become our building blocks to construct larger systems. Unlike the H-atom, where our answer was exact, we will resort to approximations for the one electron molecule. With the Born-Oppenheimer approximation, the H + Hamiltonian becomes ˆ = e e e + + H me r A r B R AB H

16 This Hamiltonian is separable in the confocal cylindrical i l coordinate system shown below, but we will not pursue the detailed solution in that system. Note however that we have the SAME symmetry y about the z-axis that we had in the H atom problem.

17 This means that we will obtain a Φ equation just like the one we saw for H, d Φ m = Φ d φ ( φ ) which h will have quantized solutions Φ(φ) = e imφ, with m = 0, ±1, ±,... As before, these values of m must reflect the quantized component of angular momentum along the z- axis. (Think about the H atom in an external electric field directed along z.) The m quantum number is given a Greek letter designation m Orbital 0 σ ±1 π ± δ Why are the values of m that are greater than 0 are doubly degenerate??

18 In complete analogy with the use of one electron ion orbitals to build up many electron atoms, these one electron molecular orbitals are our building blocks for many electron molecules. The many electron molecular wave functions are built up as products of one electron orbitals, and we then antisymmetrize the product function. We obtain Hartree-Fock limiting forms and add take correlation effects into account as for atoms. As for atoms, we must be concerned with TOTAL spin and angular momentum. The molecular cases become more complex as a result of the non-spherical molecular symmetry. Whil t t l t m l l m st b th While our prototype one electron molecule must be the general heteronuclear diatomic AB n+, we will first look at H +, and extend the treatment to H and other homonuclear diatomics. We then look at bonding in the heteronuclear diatomics.