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RHF and UHF formalisms Given a set of k orthonormal spatial orbitals (MO) {i}, i=1,...k 2k spin-orbitals: i, i=1,...,2k 2i 1 ( x ) i (r ) ( ) 2i ( x ) i (r )( ) RHF unrestricted MOs unrestricted wave-function i 1, K 1s 1s 2 s restricted MOs restricted wave-function electrons with alpha and beta spin are constrained to be described by the same spatial wavefunction Restricted wave-function for Li atom But: K1s()2s( )≠0 and K1s()2s()=0 1s() and 1s() electrons will experience different potentials so that it will be more convenient to describe the two kind of electrons by different wave-functions UHF α i α j δ ij β i β j δ ij β j S ij 1s 1s 2 s α i αβ Unrestricted wave-function for Li atom usually, the two sets of spatial orbitals use the same basis set UHF wave-functions are not eigenfunctions of S2 operator !!! spin contamination 2 c 2 2 c 4 4 c 6 6 ... 2 2 2 |2> - exact doublet state -approximately a singlet |4> - exact quartet state - approximately a doublet |6> - exact sextet state For an UHF wave-function, the expectation value of S2 is: S S 2 N 2 UHF exact N i 2 N S ij S exact S N N 2 S 2 UHF N N 2 α δ ij β β i j δ ij α β i j αβ S ij j where: 2 α i j 1 2 State S2 -> S(S+1) singlet 0.00 doublet 0.75 triplet 3.75 exact spin projection procedures (Gaussian) generate singlet eigenstates from a UHF determinat by applying projection operators that interchange the spins of the electrons Comparison of the R(O)HF and UHF formalisms R(O)HF UHF Spin-orbitals for pairs of electrons with α and β spin are constrained to have the same spatial dependence Spin-orbitals for electrons with have different spatial parts α and β spins Wavefunction is an eigenfuction of the S2 operator Wavefunction is not an eigenfuction of the S2 operator; Spin-contamination Not suitable for the calculation of spindependent properties Yields qualitatively correct spin densities EUHF ≤ ER(O)HF Different density matrices for the two sets of electrons; their sum gives the electronic density, while their difference gives the spin density For a closed-shell system in RHF formalism, the total energy and molecular orbital energies are given by (see Szabo and Ostlund, pag.83 and chapter 4 in D.B. Cook, Handbook of Computational Quantum Chemistry): N/2 E 2 H i i 1 N/2 N/2 (2J i 1 ij K ij ) j 1 N/2 i H i (2J ij K ij ) j1 Each occupied spin-orbital i contributes a term Hi to the energy Each unique pair of electrons (irrespective of their spins) in spatial orbitals i and j contributes the term Jij to the energy Each unique pair of electrons with parallel spins in spatial orbitals i and j contributes the term –Kij to the energy Or (over the spin orbitals): Each occupied spin-orbital i contributes a term Hi to the energy and every unique pair of occupied spin orbitals i and j contributes a term Jij-Kij to the energy Examples: 3 2 1 a) b) c) a) E=2H1+J11 b) E=2H1+H2+J11+2J12-K12 c) E=H1+H2+J12-K12 d) E=H1+H2+J12 e) E=H1+2H2+H3+2J12+J22+J13+2J23-K12-K13-K23 d) e) Hartree-Fock-Roothaan Equations K LCAO-MO i c {μ} – a set of known functions i=1,2,...,K i 1 The more complete set {μ}, the more accurate representation of the exact MO, the more exact the eigenfunctions of the Fock operator The problem of calculating HF MO the problem of calculating the set cμi LCAO coefficients f ( r1 ) c i ( r1 ) i c i ( r1 ) matrix equation for the cμi coefficients Multiplying by μ*(r1) on the left and integrating we get: c i * ( r1 ) f ( r1 ) ( r1 ) dr 1 i c i F ( r ) f ( r ) ( r ) dr * 1 1 1 1 - Fock matrix (KxK Hermitian matrix) S ( r ) ( r ) dr * 1 1 1 - overlap matrix (KxK Hermitian matrix) F c i i S c i , - Roothaan equations i 1, 2 ,..., K * ( r1 ) ( r1 ) dr 1 More compactly: FC=SC where c 11 c 21 C ... c K1 1 0 ... 0 c 12 ... c 22 ... ... ... cK2 ... 0 ... 2 ... ... ... 0 ... c1K c2K ... c KK -the matrix of the expansion coefficients (its columns describe the molecular orbitals) 0 0 ... K The requirement that the molecular orbitals be orthonormal in the LCAO approximation demands that: c i c j S ij The problem of finding the molecular orbitals {i} and orbital energies i involves solving the matrix equation FC=SC. For this, we need an explicit expression for the Fock matrix Charge density For a closed shell molecule, described by a single determinant wave-function N/2 ρ(r) 2 Φ a (r) 2 a The integral of this charge density is just the total number of electrons: ρ(r)dr N Inserting the molecular orbital expansion K i c i 1 into the expression for the charge density we get: N /2 ρ(r) 2 N /2 * a (r ) a (r ) 2 a μν * a N /2 * * 2 c a c a μ (r) ν (r) a Pμν μ (r) ν (r) * μν c a ( r ) c a ( r ) * Where: N /2 P 2 c a c a * - elements of the density matrix a The integral of (r) is ρ(r)dr Pμν μ (r) ν (r) dr * μν Pμν μ (r) ν (r)dr * μν Pμν S μν N μν By means of the last equation, the electronic charge distribution may be decomposed into contributions associated with the various basis functions of the LCAO expansion. Off-diagonal elements Pμ ν S μ ν -the electronic population of the atomic overlap distribution -give an indication of contributions to chemical binding when and centered on different atoms Diagonal elements Pμμ S μμ - the net electronic charge residing in orbital FC=SC. S is Hermitian and positive definite => exist the S1/2 and S-1/2 matrices with the properties: S-1/2S1/2=1 and S1/2S1/2=S Trick: multiply the HFR matrix equation from the left by S-1/2, put S-1/2 . S1/2 in front of C from the left-hand side and write S in the right-hand side as S1/2S1/2: => 1/ 2 1/ 2 1/ 2 1/ 2 S FS S C S S Cε S 1/ 2 S 1/ 2 S 1/ 2 Cε Notations: S 1/ 2 FS S 1/ 2 1/ 2 F' C C' Thus: F' C' C' ε Computational effort Time nedeed for solving the SCF equations scakes as M4 (M- # of basis functions) Accuracy Greather M → more accurate MOs and MO’s energies Complete basis set limit (HF limit): M→∞ (never reached in practice the best result that can be obtained based on a single determinantal wavefunction Population analysis - allocate the electrons in the molecule in a fractional manner, among the various parts of the molecule (atoms, bonds, basis functions) → partial atomic charges, spin density distribution, bond orders, localized MOs - Mulliken population analysis (MPA) – strongly depends on the particular basis set used Substituting the basis set expansion we get: ( r ) dr N P S ( PS ) tr PS (PS)μμ can be interpreted as the charge associated with the basis function φμ ( PS ) q M ( A ) Mulliken charge of atom A A Basis set functions are normalized Sμμ=1 Pμμ - number of electrons associated with a particular BF - net population of φμ Qμ = 2PμSμ (μ≠) overlap population - associated with two basis functions Total electronic charge in a molecule consists of two parts: K P K K Q N first term is associated with individual BF second term is associated with pairs of BF q P P S qA Z A net P A q AB A B Q K - gross population for φμ q N the net charge associated with the atom A; P is the net population of total overlap population between atoms A and B where Qμ = 2PμSμ is the overlap population between two basis functions Formaldehyde (CH2O) (aqueous solution: formol) an important precursor to many chemical compounds, especially for polymers. gas at room temperature which converts readily to a variety of derivatives. annual world production: more than 21 million tonnes. intermediate in the oxidation (or combustion) of methane as well as other carbon compounds (forest fires, automobile exhaust, tobacco smoke). can be produced in the atmosphere by the action of sunlight and oxygen on atmospheric methane and other hydrocarbons (part of smog). the first polyatomic organic molecule detected in the interstellar medium (Zuckerman, B.; Buhl, D.; Palmer, P.; Snyder, L. E., Observation of interstellar formaldehyde, Astrophys. J. 160 (1970) 485) → used to map out kinematic features of dark clouds mechanism of formation: hydrogenation of CO ice: H + CO → HCO HCO + H → H2CO (low reactivity in gas phase) Due to its widespread use, toxicity and volatility, exposure to formaldehyde is very important for human health. It is used to make the hard pill coatings that dissolve slowly and deliver a more complete dosage. Is it carcinogen? Formaldehyde Mulliken population analysis #P RHF/STO-3G scf(conventional) Iop(3/33=6) Extralinks=L316 Noraff Symm=Noint Iop(3/33=1) pop(full) Basis functions: cμi oc The summation is over occupied molecular orbitals Pμν 2 c μi c νi i1 Example P51 2(c 51 c11 c52 c12 ... c58 c18 ) S μν Pμν S μν ρ(r)dr oc oc μ ν P μν S μν N = sum over the line (or column) corresponding to the C(1s) basis function = sum over the line (or column) corresponding to the O(2px) basis function Atomic populations (AP) Total atomic charges (Q=Z-AP) 1O 8.186789 1O -0.186789 2C 5.926642 2C 0.073358 3H 4H 0.943285 0.943285 3H 4H 0.056715 0.056715 Molecular orbitals of formaldehyde (RHF/STO-3G) Excited state symmetry Formaldehyde The symmetry of the first excited state of formaldehyde (as a result of HOMOLUMO transition)