Semiempirical Molecular Orbital Theory

Semiempirical Molecular Orbital Theory Slater- determinantal wave func:ons and artree- Fock theory Video III.iv

Constructing a 1-Electron Wave Function The units of the wave function are such that its square is electron per volume. As electrons are quantum particles with non-point distributions, sometimes we say density or probability density instead of electron per volume (especially when there is more than one electron, since they are indistinguishable as quantum particles) For instance, a valid wave function in cartesian coordinates for one electron might be: "( x, y,z; Z) = 2Z 5 / 2 81 # % ' 6 $ Z x 2 + y 2 + z 2 & ( * ye $Z x 2 +y 2 +z 2 / 3 ) normalization factor radial phase factor cartesian directionality (if any) ensures square integrability

Constructing a 1-Electron Wave Function A valid wave function in cartesian coordinates for one electron might be: "( x, y,z; Z) = 2Z 5 / 2 81 # % ' 6 $ Z x 2 + y 2 + z 2 & ( * ye $Z x 2 +y 2 +z 2 / 3 ) normalization factor radial phase factor cartesian directionality (if any) ensures square integrability x 1 x x 2 P y 1 y y 2 = x 2 y 2 z 2 φ 2 dx dy dz x 1 y 1 z 1 z 1 z z 2 This permits us to compute the probability of finding the electron within a particular cartesian volume element (normalization factors are determined by requiring that P = 1 when all limits are infinite, i.e., integration over all space)

Constructing a 1-Electron Wave Function To permit additional flexibility, we may take our wave function to be a linear combination of some set of common basis functions, e.g., atomic orbitals (LCAO). Thus ( ) = a i ϕ( r) φ r N i=1 For example, consider the wave function for an electron in a C bond. It could be represented by s and p functions on the atomic positions, or s functions along the bond axis, or any other fashion convenient. C

Constructing a 1-Electron Wave Function To optimize the coefficients in our LCAO expansion, we use the variational principle, which says that E = a i ϕ i a j ϕ j dr i j a i ϕ i a j ϕ j dr i j = ij ij a i a j a i a j ϕ i ϕ j dr ϕ i ϕ j dr = ij ij a i a j a i a j ij S ij which is minimized by one root E of the secular equation (the other roots are excited states) each value of E that satisfies the secular equation determines all of the a i values and thus the shape of the molecular orbital wave function (MO) 11 ES 11 12 ES 12 1N ES 1N 21 ES 21 22 ES 22 2N ES 2 N = 0 N1 ES N1 N2 ES N2 NN ES NN

What Are These Integrals? The electronic amiltonian includes kinetic energy, nuclear attraction, and, if there is more than one electron, electron-electron repulsion ij = ϕ i 1 nuclei electrons 2 Z ϕ j ϕ k φ 2 i ϕ j + ϕ i 2 k r k n r n ϕ j The final term is problematic. Solving for all electrons at once is a many-body problem that has not been solved even for classical particles. An approximation is to ignore the correlated motion of the electrons, and treat each electron as independent, but even then, if each MO depends on all of the other MOs, how can we determine even one of them? The artree-fock approach accomplishes this for a many-electron wave function expressed as an antisymmetrized product of one-electron MOs (a so-called Slater determinant)

Choose a basis set Choose a molecular geometry q (0) Compute and store all overlap, one-electron, and two-electron integrals Guess initial density matrix P (0) The artree- Fock procedure Construct and solve artree- Fock secular equation Replace P (n 1) with P (n) Choose new geometry according to optimization algorithm no no Optimize molecular geometry? yes Does the current geometry satisfy the optimization criteria? yes Output data for optimized geometry Construct density matrix from occupied MOs Is new density matrix P (n) sufficiently similar to old density matrix P (n 1)? no yes Output data for unoptimized geometry F 11 ES 11 F 12 ES 12 F 1N ES 1N F 21 ES 21 F 22 ES 22 F 2N ES 2N F N1 ES N1 F N2 ES N2 F NN ES NN nuclei F µν = µ 1 2 2 ν Z k µ 1 ν r k + P λσ λσ ( µν λσ) = φ µ ( 1)φ ν 1 ( ) 1 k ( µν λσ) 1 2 ( µλ νσ) occupied P λσ = 2 a λi φ r λ ( 2)φ σ ( 2)dr ( 1)dr 2 12 i = 0 a σi ( ) One MO per root E

Cement Your Understanding F µν = µ 1 2 2 ν Z k µ 1 ν r k + P λσ λσ nuclei k ( µν λσ) 1 2 ( µλ νσ) Try to write out, neatly and carefully, in full mathematical notation, every generic integral in the Fock matrix for Slater-type orbital basis functions with all functions and operators expressed in Cartesian coordinates (which is how they are most typically evaluated in actual practice)

Semiempirical Molecular Orbital Theory Semiempirical simplifica:ons of the Fock matrix (early) Video III.v

nuclei F µν = µ 1 2 2 ν Z k µ 1 ν r k + P λσ λσ k ( µν λσ) 1 2 ( µλ νσ) occupied P λσ = 2 a λi φ i = a µi ϕ µ µ i a σi F 11 ES 11 F 12 ES 12 F 1N ES 1N F 21 ES 21 F 22 ES 22 F 2N ES 2N = 0 F N1 ES N1 F N2 ES N2 F NN ES NN Consider only first- and second- row atoms; consider a minimal basis set ( µν λσ) = ϕ µ ( 1)ϕ ν 1 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2

nuclei F µν = µ 1 2 2 ν Z k µ 1 ν r k + P λσ λσ ( µν λσ) = ϕ µ ( 1)ϕ ν 1 k ( µν λσ) 1 2 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2 Things to consider in a semiempirical model: ( µλ νσ) occupied P λσ = 2 a λi φ i = a µi ϕ µ µ i a σi 1) What shall I choose for a basis set? 2) What shall I do for overlap integrals (in the secular determinant)? 3) What shall I do for Fock matrix elements? 4) As part of (3), what shall I do about electron repulsion integrals? Essen:ally all semiempirical models adopt a minimal basis set. The AO basis func:ons cover only the valence s, p, d, and f orbitals (p, d, and f only as needed), one func:on per orbital (so, 1 s, 3 p, 5d, and 7f orbitals). The basis func:ons themselves are taken to be Slater orbitals. ( ϕ STO ( r,θ,φ;ζ,n,l,m) = 2ζ ) n +1/ 2 ( 2n)! [ ] 1/ 2 rn 1 e ζr Y l ( ) m θ,φ

nuclei F µν = µ 1 2 2 ν Z k µ 1 ν r k + P λσ λσ ( µν λσ) = ϕ µ ( 1)ϕ ν 1 k ( µν λσ) 1 2 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2 ( µλ νσ) occupied P λσ = 2 a λi φ i = a µi ϕ µ µ i a σi General choice for overlap integrals, S µν : The overlap between any two STOs can be computed analy:cally (i.e., it is a simple arithme:c func:on of posi:on and values of ζ, n, l, and m). Overlap integrals find use in compu:ng some Fock matrix elements. owever, when forming the secular determinant, the simplifica:on S µν = δ µν is adopted. ϕ STO ( ) n +1/ 2 ( r,θ,φ;ζ,n,l,m) = 2ζ ( 2n)! [ ] 1/ 2 rn 1 e ζr Y l ( ) m θ,φ

nuclei F µν = µ 1 2 2 ν Z k µ 1 ν r k + P λσ λσ ( µν λσ) = ϕ µ ( 1)ϕ ν 1 k ( µν λσ) 1 2 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2 ( µλ νσ) occupied P λσ = 2 a λi φ i = a µi ϕ µ µ i a σi There are 3 types of Fock matrix integrals, F µν : A Fock matrix element can be diagonal (i.e., µ = ν), or not. When the matrix element is off- diagonal, it can be monatomic (i.e., µ ν, but µ and ν are on the same atom), or it can be diatomic (the only remaining possibility) As every Fock matrix element includes ERIs, first let s consider the ERIs (we ll get to one- electron integrals later)

COMPLETE NEGLECT OF DIFFERENTIAL OVERLAP Reducing the N 4 scaling problem: ( µν λσ) = ϕ µ ( 1)ϕ ν 1 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2 CNDO: ( µν λσ) = δ µν δ λσ ( µµ λλ) only ERIs not equal to zero are Coulomb integrals between 2 AO basis func:ons, which may or may not be on the same atom ( µµ λλ) = γ AB γ is a number or func:on that depends only on the atoms A and B on which µ and λ are found, respec:vely. γ AA = IP A EA A interac:on between electrons on same atom taken as difference between ioniza:on poten:al and electron affinity γ AB = γ AA + γ BB ( ) 2 + r AB γ AA + γ BB func:on above has proper limits at r AB = 0 (since then A must be B, the same atom) and at r AB very large (just 1/r AB, which is Coulomb s law in atomic units) M + M IP EA M + M one extra e-e repulsion on atom M

Semiempirical Molecular Orbital Theory Semiempirical simplifica:ons of the Fock matrix (subsequent) Video III.vi

INTERMEDIATE NEGLECT OF DIFFERENTIAL OVERLAP Reducing the N 4 scaling problem: ( µν λσ) = ϕ µ ( 1)ϕ ν 1 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2 INDO: ( µν λσ) = δ µν δ λσ ( µµ λλ) s:ll taken to be true unless µ and λ are on the same atom ( ss ss) = G ss ( ss pp) = G sp ( pp pp) = G pp ( pp pʹ pʹ ) = G pp ʹ ( sp sp) = L sp For an atom with only s and p valence func:ons, 5 unique integrals are kept (they are simply values, in energy units, that can be determined from spectroscopy or taken as free parameters). Note that p refers to a p orbital different from a reference one, e.g., ( pp p p ) might be (p x p x p y p y ), i.e., the through- space repulsive interac:on between electrons in two perpendicular p orbitals This extra flexibility in the atomic ERIs tends to improve rela:ve energies for electronic states that differ by occupa:on of orbitals on single- centers. That is, for instance, excited states that are well characterized as one- electron local excita:ons. INDO s:ll finds substan:al use for spectroscopy.

NEGLECT OF DIATOMIC DIFFERENTIAL OVERLAP Reducing the N 4 scaling problem: ( µν λσ) = ϕ µ ( 1)ϕ ν 1 ( ) 1 r 12 ϕ λ 2 ( )ϕ σ ( 2)dr 1 dr 2 NDDO: ( µν λσ) = δ A µ X ν δ Bλ Y σ ( µν λσ) so, the only integrals that survive are those where both µ and ν are on the same atom, ( ss ss) = G ss ( ss pp) = G sp ( pp pp) = G pp ( pp pʹ pʹ ) = G pp ʹ ( sp sp) = L sp and λ and σ are on the same atom, but neither the two atoms need to be the same, nor do the two func:ons on the individual atoms need to be the same. For monatomic ERIs, the INDO- like values are retained. For diatomic ERIs, there are 100 possible integrals for s and p basis func:ons (try and verify that ) The diatomic integrals are not easily evaluated for Slater type orbitals. Instead, NDDO methods model the integrals as classical mul:pole interac:ons between the two atoms. Whether the mul:pole is a point charge (ss), a dipole (sp), or a quadrupole (pp or pp ) depends on the nature of the orbitals, and the orienta:on of the higher mul:poles does, too. The magnitudes of the dipoles and quadrupoles depend on the exponents of the Slater basis func:ons.

ONE- ELECTRON INTEGRALS µ 1 nuclei 2 2 ν Z k µ 1 ν r k Consider diagonal term, µ = ν: k µ 1 2 2 µ Z k µ µ 1 nuclei µ Z r k µ 1 µ k µ r k nuclei = U µ Z k k k µ ( µµ s k s k ) k k µ where U is a parameter that should be nearly equal to the ioniza:on poten:al of an electron in atomic orbital µ and the alrac:on to a nucleus at the posi:ons of other atoms k is Z k :mes the repulsion with an electron in an s orbital on the same atom.

ONE- ELECTRON INTEGRALS µ 1 nuclei 2 2 ν Z k µ 1 ν r k k Consider off- diagonal term, µ ν, both on same atom µ 1 2 2 ν Z k µ µ 1 nuclei ν Z r k µ 1 ν k µ r k nuclei = Z k k k µ ( µν s k s k ) k k µ where the first two terms are zero for s and p orbitals by symmetry considera:ons and the alrac:on to a nucleus at the posi:ons of other atoms k is Z k :mes the repulsion with an electron in an s orbital on the same atom.

ONE- ELECTRON INTEGRALS µ 1 nuclei 2 2 ν Z k µ 1 ν r k k Consider off- diagonal term, µ ν, each on a different atom ( = β + β µ ν) S 2 µν where β is our old friend from ückel theory, the resonance integral (a numerical parameter) and S is the overlap matrix computed from the Slater type orbitals. Note that the β values depend on both the atomic number and the orbital (i.e., carbon has a different β value for an s orbital than for a p orbital) What would a table of parameters look like for an NDDO model?

Semiempirical Molecular Orbital Theory Semiempirical parameteriza:ons and modern usage Video III.vii

Big Personali:es John Pople Michael Dewar

FINAL MOLECULAR- MECANICS- LIKE TOUCUP First adopted in MNDO, later in AM1 and PM3, etc. V N ( A,B) = Z A Z ( B s A s A s B s ) B 4 i=1 + Z AZ B a r A,i e b A,i r AB c A,i AB ( ) 2 + a B,i e b B,i ( r AB c B,i ) 2 where the first term says that nuclei repel each other propor:onal to the degree that their s electrons repel one another, but the following sum effec:vely creates gaussian shaped ripples, posi:ve or nega:ve, in the poten:al energy surface surrounding the atoms. Such an approach permits a fine tuning of atomic separa:ons that must be considered rather ad hoc, but not necessarily inconsistent with semiempirical principles. (Later extended by Jorgensen and co- workers using Pairwise Distance Directed Gaussians (PDDG) for MNDO and PM3.)

Chemical Intui:on vs Maths Z = Observables i Occurrences j ( calc i, j expt i, j ) 2 2 w i 1/2 Michael Dewar MNDO AM1 AMPAC Jimmy Stewart MNDO AM1 PM3 MOPAC

Errors in Computed eats of Forma:on (kcal/mol)

MODERN USAGE? SPECIFIC RANGE PARAMETERS Parameter AM1 AM1-SRP C U s 52.03 49.85 C O O + + C 2 2 O U p 39.61 40.34! s 15.72 16.91! p 7.72 9.19 O U s 97.83 99.18 U p 78.26 80.76! s 29.27 29.00! p 29.27 29.25 Source "E rxn D e (C ) D e ( ) AM1 AM1-SRP Expt. 28.0 4.9 5.1 81.4 104.4 104.4 109.4 109.4 109.5