3: Many electrons. Orbital symmetries. l =2 1. m l

Transcription

1 3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience a cylindrically symmetric electric field. This leads to a Stark splitting of the atomic l states into components with m l = 0 and m l = ±1, etc., where m l is the component of l along the symmetry axis of the field (the bond axis). 0 l = m l

2 Cont. The symmetries of the orbitals in diatomic molecules are labelled by the magnitude of the component of the angular momentum along the internuclear axis, λ. Looking down the symmetry axis λ = 0 (σ) λ = 1 (π) λ = 2 (δ) (cf. particle on a ring - see Problem sheet 2, question 3)

3 Cont. For H + 2 Ψ + e is the 1σ g orbital + g Ψ e is the 1σ u orbital + - u The 1 indicates that Ψ + e is the lowest orbital of that symmetry. The g and u symmetry labels refer to the inversion symmetry, Î, with respect to the molecular frame ÎΨ e (x,y,z) = ±Ψ e ( x, y, z) u - + g

4 The orbital approximation Treat many electron molecules in same way as many electron atoms. Define a molecular orbital (MO) as a one-electron wavefunction in a molecule. In some cases these MO s may be constructed from eigenfunctions of some effective one-electron Hamiltonian. Write the total wavefunction in the many electron case as a product of one electron MOs Ψ tot (r 1,r 2, r N ) = (φ a (r 1 )φ b (r 2 ) φ n (r N )) space [Φ(1)Φ(2) Φ(N)] spin. Here, the φ i are the molecular orbitals (e.g., the 1σ g orbital of H + 2 just discussed). The total wavefunction is associated with a particular electron configuration. Ψ tot is a product of a spatial wavefunction (which describes where the electrons are) and a spin part (which tells us whether the spin of each electron is up or down ).

5 Pauli exclusion principle The total wavefunction must be constructed so that it satisfies the Pauli exclusion principle. It must be antisymmetric with respect to exchange of any two identical electrons. For example, the ground state of the H 2 molecule has an electron configuration 1σ 2 g. The Pauli principle can only be satisfied if the electron spins are paired. The total wavefunction can be written, approximately Ψ tot (r 1,r 2 ) = (φ 1σg (r 1 )φ 1σg (r 2 )) 1 2 [α(1)β(2) α(2)β(1)]. In this case the spatial wavefunction is symmetric, and the spin function is antisymmetric: Ψ tot (r 2,r 1 ) = (φ 1σg (r 2 )φ 1σg (r 1 )) 1 2 [α(2)β(1) α(1)β(2)] = Ψ space Φ spin = Ψ tot (r 1,r 2 ) In general, a total (anti-symmetrized) wavefunction satisfying the Pauli Principle can be constructed using a Slater determinant (see Atkins MQM), which generates an antisymmetric sum of product terms in Ψ tot, rather than a single product.

6 Labelling of molecular states Recall that molecular orbitals are labelled according to λ. In the many electron case, different electronic states are generated according to how the electrons fill up the available orbitals. Molecular states are labelled with a term symbol: 2S+1 Λ. Λ is the magnitude of the component of the total electronic orbital angular momentum along the internuclear axis. The total component of the orbital angular momentum is obtained from the sum Λ = λ 1 +λ 2 +λ 3 + S is the total spin angular momentum quantum number, obtained from the vector sum S = s 1 +s 2 +s 3 + Note that the spin angular momentum is not coupled to the internuclear axis (i.e. is not affected so strongly by the electric field gradient) in the molecule.

7 Predictions of the orbital approximation Ground electronic states Molecule Config n Term No. of R e /Å D e /kjmol 1 Symbol bonding e s H + 2 (1σ g ) 1 2 Σ + g H 2 (1σ g ) 2 1 Σ + g He + 2 (1σ g ) 2 (1σ u ) 1 2 Σ + u He 2 (1σ g ) 2 (1σ u ) 2 1 Σ + g 0 see below MO picture provides qualitative rationale for experimental observations. Note: The electronic states are also labelled according to their inversion symmetry (i.e. g/u), and their symmetry (+/ ) with respect to reflection in the molecular plane see Problem sheet 2, question 3). σ orbitals, and therefore the states they lead to when occupied, are symmetric with respect to this operation. Doubly occupied orbitals are also totally symmetric (i.e. (g g) = g, (u u) = g, (+ +) = +, and ( ) = +).

8 Nature of the orbital approximation Does the orbital approximation for Ψ tot satisfy the Schrödinger equation? Look at H 2 ground state again ĤΨ tot = ( ĥ 1 +ĥ ) Ψ space Φ spin. r 12 ĥ i is the Hamiltonian for electron i in H + 2, and 1/r 12 is the electron-electron Coulomb repulsion. The Hamiltonian does not operate directly on the spin function, so we can write ĤΨ space = ( ĥ 1 +ĥ ) φ 1σg (r 1 )φ 1σg (r 2 ). r 12 This yields ĤΨ space = (ǫ 1σg +ǫ 1σg )Ψ space + 1 r 12 Ψ space. The last term is not a constant Ψ space.

9 Cont. To allow for this electron-electron repulsion term Ψ space must depend on r 12, the electronelectron separation. The electron positions described by Ψ space must be correlated - the orbital approximation does not allow for this electron correlation Note: Some improvements can be made by optimizing the orbitals so as to minimize the average inter-electron repulsion using the variation principle. This leads to the self-consistent field or Hartree-Fock orbitals (see below). However, these procedures do not eliminate problems associated with the neglect of electron correlation.

10 Consequences of the orbital approximation A. Dispersion forces He 2 is predicted to be unbound because anti-bonding orbitals are more antibonding than bonding orbitals are bonding. u He(1s) He(1s) g He 2 In reality the electron positions are correlated, leading to the van der Waals (dispersion) interaction, V(R) 1/r 6. Leads to a binding energy of ǫ/k B 11K. Within the orbital approximation, for example the ground state of H 2 is represented P(r 1,r 2 ) = Ψ e 2 = P 1σg (r 1 )P 1σg (r 2 ) i.e. a product of probabilities. The electron positions are intrinsically independent.

11 B. Dissociation Examine the large R behaviour of the electron density in the molecular orbital wavefunction for H 2 φ 1σg (r 1 )φ 1σg (r 2 ) 2 R = 1s A (r 1 ) 2 1s B (r 2 ) 2 + 1s B (r 1 ) 2 1s A (r 2 ) 2 + 1s A (r 1 ) 2 1s A (r 2 ) 2 + 1s B (r 1 ) 2 1s B (r 2 ) 2 (note that the cross terms vanish as R ) i.e. dissociation products are H + H + H + + H, and the estimate of the dissociation energy is very poor.

12 Valence bond description An alternative is to write a valence bond wavefunctions for the ground electronic state of H 2 Ψ VB (r 1,r 2 ) = N [1s A (r 1 )1s B (r 2 )+1s A (r 2 )1s B (r 1 )] This is not an orbital approximation. This completely removes the ionic terms in the wavefunction, leading to a more accurate description at the dissociation limit. However, the description of the molecule is poor for the same reason: the valence bond wavefunction is over-correlated.

13 Cont. A better approach is to mix the MO and VB wavefunctions Ψ e = N(λ)[Ψ MO +λψ VB ], where λ is optimized, using the variation principle, at each value of R. V( R) H 2 exact LCAO VB CI/MO+VB 1 + X( g ) H(1s) + H(1s) In this way the optimum mix of covalent and ionic character can be obtained at each R. R

14 Configuration interaction The same wavefunction is obtained using the MO wavefunction with configuration interaction. The configurations (1σ g ) 2 and (1σ u ) 2 both lead to electronic states of 1 Σ + g symmetry. Ψ 1 = φ 1σg (r 1 )φ 1σg (r 2 ) Ψ 2 = φ 1σu (r 1 )φ 1σu (r 2 ) Both of these electronic states correlate to the same H + H + H + + H products. 1 + g H 2 LCAO CI V( R) H H 1 + X( g ) H(1s) + H(1s) Better trial wavefunctions for these two 1 Σ + g states are obtained by mixing the two configurations Ψ ± = Ψ 1 ±λ Ψ 2. The optimized ground state wavefunction is the same as that obtained using the mixed valence bond/mo wavefunction. More details can be found in Green QM2. R

15 C. Configurations and terms The orbital approximation predicts that all terms arising from a given configuration will be degenerate. In reality, they can be split into different terms. The energy of a configuration is not obtained simply as the sum of the occupied orbital energies! Consider a (π) 2 configuration (see later for more discussion of B 2 and O 2 ). The π orbitals are two-fold degenerate, corresponding to an electron rotating about the bond axis in opposite directions, leading to m l = ±1. In reality, the above arrangements of the two electrons have different energies, because of differences in the inter-electron repulsion.

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17 4: Applications of the variation principle Secular equations Provides a general route to approximate molecular wavefunctions and orbital energies. Let Ψ be some trial (approximate) wavefunction. Then, according to the variation principle E E = Ψ Ĥ Ψ Ψ Ψ E 0, where E 0 is the true ground state. Apply this to LCAO molecular orbitals Ψ = N m c n φ n, n=1 where N is a normalization constant, and c n is the (real) coefficient for the nth basis function, φ n. Assume also that the basis functions, φ n, are normalized. For example, for the first row diatomics the φ n might be the basis set comprised of the 1s, 2s, 2p x, etc. orbitals on each atom.

18 Introduce simplifying notation ( )( Ψ Ψ = c n φ n n m where the overlap integral is defined c m φ m ) dτ = n,m c n c m S nm, Cont. S nm = φ n φ m dτ. Note that S nm = S mn. The numerator can be written ( ( ) Ψ Ĥ Ψ = c n φ n )Ĥ c m φ m dτ = n m n,m where the matrix elements are defined H nm = φ n Ĥφ m dτ. c n c m H nm, The Hermitian properties of Ĥ ensure that H nm = H mn.

19 Cont. To find the best estimate of E 0, minimize E with respect to the coefficients, c n. E c k = 1 Ψ Ĥ Ψ Ψ Ψ c k Ψ Ĥ Ψ Ψ Ψ 2 Ψ Ψ c k = 0 or Ψ Ĥ Ψ c k Ψ Ĥ Ψ Ψ Ψ Ψ Ψ c k = 0. The partial derivatives can be written explicitly Ψ Ĥ Ψ c k = 2 m c m H km and Ψ Ψ c k = 2 m c m S km, where the factor of 2 arises from the double summations in Ψ Ψ and Ψ Ĥ Ψ.

20 Cont. Therefore (H km ES km )c m = 0 for all k = 1 m. m In matrix notation this can be written (H ES)c = 0, which are known as the secular equations. The non-trivial solutions are given by det H ES = 0. Solution yields a polynomial with m roots, or eigenvalues, i.e. m MO energies, E, the lowest of which is the best estimate of the ground state energy. Once the eigenvalues, E, are known, the coefficients, c m, can be calculated by back-substitution into the secular equations. For the lowest root one obtains the best estimate of Ψ 0.

21 Illustration for two orbitals Consider a heteronuclear diatomic molecule, e.g., a simple model of the HOMO of LiH Ψ c 1 φ Li +c 2 φ H, where φ Li could be the 2s orbital on Li, and φ H could be the 1s orbital on the H atom. The secular equation reduces to H 11 E H 12 ES 12 H 21 ES 21 H 22 E c 1 c 2 = 0 0 H 11 = α A and H 22 = α B are approximately the energies of the 2s orbital on Li ( 5eV = I.P. of Li) and the 1s orbital on H ( 13eV = I.P. of H). H 12 = H 21 = β 2eV at R e, known as the resonance integral, is the energy associated with the overlap density. S 12 = S 21 = S 0.4 at R e. Note that S 11 = S 22 = 1

22 Solving to get the energy levels. Multiplying out the secular determinant gives (α A E)(α B E) (β ES) 2 = 0, the solution of which is E = α A +α B 2βS ± (α A +α B 2βS) 2 4(1 S 2 )(α A α B β 2 ) 2(1 S 2 ) (1) To progress further, either 1. Stick the numbers into a computer 2. Set S = 0 (provides a useful guide to the general behaviour see below) 3. Set α A = α B = α and check that we get the homonuclear H + 2 case.

23 Check for the H + 2 case... E ± = α±β 1±S. This equation agrees with that for H + 2, when explicit results for α (= H AA) and β (= H AB ) are substituted.... and solving to get the wavefunctions. To get the orbitals substitute back into the secular equations. For example, for the ground state (Ψ + ) ( α α+β ) ( c A + β (α+β)s 1+S 1+S ) c B = 0, c A = c B. Normalization, as before (see p11), leads to c A = c B = 1 [2(1+S)] 1/2, as produced before using symmetry arguments. The same procedure can be used for the other root (i.e. Ψ ).

24 The two orbital case without overlap Using equation 1, setting S = 0 ( E = 1 2 α A +α B ± ) (α A +α B ) 2 4(α A α B β 2 ). When α A = α B = α E - = E ± = α±β E + = When α A α B E ± = α A +α B 2 ± α A α B 2 ( 1+ 4β 2 (α A α B ) 2 ) 1/2 B E - If α A α B = 2 β E +

25 Energies For small β/ α A α B the energies E ± can be approximated E ± = α A + β2 α A α B + α B β2 α A α B + Plot the MO energies versus (α A α B ) at fixed β. E - E/ A B extent of stabilization of lower orbital E + =0 This reveals that atomic orbitals of very different energies do not mix.

26 Wavefunctions For α A α B (so that orbital of A is lower than that of B), with and Ψ + = c + A φ A +c + B φ B then Ψ = c A φ A +c B φ B, c + A c + B = 2 +4β 2 + 2β with = α A α B. For LiH /β c /c + + A B c /c - - A B 3 A B)/