SPTP CHE 596F
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This 58 page Class Notes was uploaded by Mr. Lilian Johns on Thursday October 15, 2015. The Class Notes belongs to CHE 596F at North Carolina State University taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/223787/che-596f-north-carolina-state-university in Chemical Engineering at North Carolina State University.
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Date Created: 10/15/15
Molecular Structure BornOppenheimer approximation LCAOMO application to H2 The potential energy surface Hartree Fock theory MOS for diatomic molecules Post Hartree Fock Configuration Interaction BornOppenheimer approximation Molecular motion includes the motion of both nuclei and electrons The time scale of the motion of the nuclei is orders of magnitude slower than electron motion due to the difference in mass me 9109x1031kg mp 1672x 10 27 kg The potential energy surface The idea that nuclear and electronic motions can be separated implies that there is a nuclear potential energy surface UR where R is the internuclear distance The potential surface has an equilibrium geometry Re and well depth De Hydrogen molecule ion wave functions We use this example because it can be solved exactly The spatial wavefunction on each of two H atoms forms linear combinations e TH1SA rA I B LIIFHSB 1 R A B The atomic wave functions form linear combinations to make molecular orbital wave functions LPi lsAi 15B The hamiltonian for H2 The potential involves three particles one electrons and two protons ln atomic units it is given by L i i V mpg R The hamiltonian includes the kinetic energy terms for the electron only since the BornOppenheimer approximation allows separation of nuclear and electronic motion The internuclear distance R is fixed and the nuclear kinetic energy is zero H 2V V Setting up the energy calculation for H2 The average energy is obtained by evaluating the expectation value I VHWT E 5U5Udt The denominator gives the required normalization 50 sud 1 1s11sj31sA lsBdI I lsglsAdr f lsglsBdrf lsglsBdrf lsjglsAdr 1S1Swherefls41SAd C 1andf 132133611 S Significance of the overlap integral The wave functions 1sA and 1sB are not orthogonal since they are centered on different nuclei The overlap integral 8 is a function of the internuclear distance A B O O O jrlap region Normalized LCAO wave functions The LCAO wave functions for the H2 molecule ion are R Why 1sB and 5 1sA 1SB HGS These wave functions are orthogonal and normalized Energy levels in H2 Explicit substitution of the hamiltonian gives 1 1 1 1 E 50 H f dI l av ZVZ a EFlfldr f1sj4 Vz ls3dt f lsquot3 V2 r Z BlsAdI Energy levels in H2 f 132Elsr IBlsAdtf 1S4E1s 1S3d t f lsgEls lsBdtf lsgEls B13Adt Since 1 2 11 These are 39ust V S E ls J 2 FA A 15 A hydrogen atom Schr dinger l A 2 equations 2V r3lsB ElslsB Energy levels in H2 Els1S Jr ls61 B15Adr f ls BlsBdT E1s1Sf Isu sl f 1S B1SA0LC To further evaluate these twocenter integrals we define the Coulomb integral J f 1SAr13121SAd C Eris1 d1 and the exchange integral 1513 K f 133 3lsAdII i BAdI Energy levels in H2 2 5 HTW E1321S2J K 15 f 5U SUdT 21S E39E1s I E 2 T HT dT Z E1S21 S 2J K sU svm 21 5 J K E E1S 1S Diagram of H2 energy levels J K 1 S antibonding E15 E13 bonding J K 1 S Note that the antibonding level is more destabilizing than the bonding level is stabilizing Potential energy surface for H2 Using elliptical integrals that S J and K integrals can be solved analytically to yield the following PES SR eR1 R JR e2R1 KR eR1R Anti bonding W 3 4567 Mao MO treatment of H2 antibonding S I l I t t t t t I bonding The two electrons must have opposite spinsor and 3 The wave function must be antisymmetric with respect to electron exchange sum sablt1gtsvblt2gtoclt1gtr3lt2gt alt2gtBlt1gt Spatial part Spin part antisymmetric t t t I l I Is I 1 l I I Application to diatomic molecules Li2 B62 B2 C2 N2 02 F2 ll 4 14 if till rill iii 26 4 16 1 4 H H 4 Considering only valence electrons we H H can fill the molecular orbitals of diatomics Linear combinations of 2s 2pZ give s orbitals Linear combinations of 2pxy give p orbitals The relative energy ordering depends on the number of electrons in occupied orbitals The HartreeFock Method Introduction An approximate method for manyelectron atoms was first proposed by Hartree In this method the atomic wave function is a product of oneelectron wave functions This an extension of what we have seen for helium LP quot1 2 3 N 1 1 2 2 3 3 N N In this equation the ri are the positional coordinates and a spin coordinate for each electron ri xiyizimi As we have seen for the hydrogen atom spherical polar coordinates work better than Cartesian coordinates The spin coordinate can be spin up or or spin down 3 The atomic electronic hamiltonian We reintroduce the hamiltonian for He to introduce commonly used abbreviations e L2 V2 39 Uee r12 2m 1 sz 72 2 2 2 m 2 6 V1 262 V2 T Z 71 2 72 The kinetic energy operator is the e 139 2m 2 1 sum of the kinetic terms for all of the VM 2 Z 2 electrons The external potential is the field of nuclei Coulomb attraction felt by the electrons The nuclear hamiltonian The BornOppenheimer approximation states that electronic and nuclear motion can be separated because of the large difference in mass 1 proton has 1836 times the mass of an electron Thus the nuclear kinetic energy and repulsion terms are not included However it is important to understand that the nuclear terms are used for translational rotational and vibrational motion Vibrations are of particular relevance since the quantum chemical calculation of vibrational frequencies is the basis for parameterization of classical molecular force fields 2 2 Tm 2 Kinetic energy term 2 Zje Um R Nuclear repulsion energy 139 The effective hamiltonian Using these approximations we can write the hamiltonian of an atom as H61Te V ext U66 The first two terms alone comprise a set of N hydrogen atom calculations which can be solved exactly 7W ZVfzl2hi ext The total hamiltonian in this nomenclature is Helzzhi Uee Note that the inclusion of the electronelectron repulsion term Uee makes it impossible to solve the Schrodinger equation using this hamiltonian The Hartree approximation The method is to solve the Schrodinger equation for each individual electron in the field of all of the other electrons If we assume that we know the individual wave functions ltgtir for each electron we can calculate the electron density according to pi IMF The total electron density is pm E p E litFir However the kth electron does not interact with itself so Must subtract the density of rk from the total pa pm pk 2 pi ltIgtkr2 ltgtir2 i k l2 Limitations of the Hartree approximation The Hartree approximation works well for atoms However the form of the wave function is not correct and the method fails for molecules One property of the wave function is that is must change sign when any two electrons are interchanged Around 1930 both Fock and Slater proposed to fix the problem with Hartree model by introducing a wave function that is antisymmetric with respect to electron exchange 1 r Pr1r2r3rN 3 ilrrN 2rNi3rN NrN The wave function is the determinant of the matrix Overview of the Hartree method The kth electron is now treated as a point charge in the field of all of the other electrons This procedure takes the manyelectron problem and simplifies it to many one electron problems Oneelectron system with remaining electrons represented by an average charge density Mean field approach Manyelectron system All electronelectron repulsion is included explicitly The Hartree procedure The interaction of the point charge with the electron density is 1 gkr 2 Ir rlldr Making this approximation we can write N Uee z gir and the hamiltonian Hel now consists of oneelectron operators N He z hi 8239 The manyelectron Schrodinger equation can now be Solved as N oneelectron Schrodinger equations 1 hi gili 8i i The Coulomb integral The set of oneelectron Schrodinger equations can be solved iteratively to find the best energy According to the variational principle the true energy will always be lower than the energy calculated using this method However this method counts the interaction between each pair of electrons twice Therefore we cannot simply add the individual oneelectron energies 8i to obtain the total energy E We correct for this using the Coulomb integral between each pair of electrons JU JUMWW ij lrz rll 1 2 The exchange integral The determinantal wave function adds a new term to the Hartree energy the exchange term K N 1 N N E 2139 8i 121 We compare the Coulomb and exchange integrals below f f ltlgtr1ltlgtir1m fr2 jrzdmdrz f f ltlgtr1ltlgtjrJmh z z mda Notice that the difference is a rather subtle swap of the indices i and j This small change has a profound effect on the contribution calculated by the integral The Fock equations The procedure leads to the Fock equations fr1 irl 8i irl The Fock operator is a oneelectron operator fr1 W1 i m1 km The choice of electron 1 is arbitrary These equations can written for all of the electrons The Coulomb and exchange terms are mm m ar2lzmdrz WW mm wwmdrz Application of the Variational Method The linear combination of atomic orbitals LCAO Calculations of the energy and properties of molecules requires hydrogenlike wave functions on each of the nuclei The HartreeFock method begins with assumption that molecular orbitals can be formed as a linear combination of atomic orbitals N bi 211 Crux The basis functions x are hydrogenlike atomic orbitals that have been optimized by a variational procedure The HF procedure is a variational procedure to minimize the coefficients Cm Note that we use the index u for atomic orbitals and i orj for molecular orbitals Common types of atomic orbitals Slatertype orbitals STOs x 0C 6 The STOs are like hydrogen atom wave functions The problem with STOs arises in multicenter integrals The Coulomb and exchange integrals involve electrons on different nuclei and so the distance r has a different origin Gaussiantype orbitals GTOs W2 XHOC 9 Gaussian orbitals can be used to mimic the shape of exponentials ie the form of the solutions for the hydrogen atom Multicenter Gaussian integrals can be solved analytically STOs vs GTOs GTOs are mathematically easy to work with Normalized Amplitude r atomic units STOs vs GTOs GTOs are mathematically easy to work with but the shape of a Gaussian is not that similar to that of an exponential Normalized Amplitude r atomic units STOs vs GTOs Therefore linear combinations of Gaussians are used to imitate the shape of an exponential Shown is a representation of the 3Gaussian model of a STO expr Sum STOSG Gaussian components O2exp O16 r2 04exp 067 r2 O3exp 25 r2 Normalized Amplitude r atomic units Doublezeta basis sets Since the remaining atoms have a different exponential dependence than hydrogen it is often convenient to include more parameters XH Dude Car Dilbe Cbr The second exponential is a diffuse function It accounts for properties of a valence electrons involved in bonding When GTOs are used there are always multiple Gaussians required because the shape of Gaussians must be matched as closely as possible to that of exponentials In a doublezeta basis there may be up to 3 Gaussians used to represent the first exponent Q8 and 1 for the second exponent Qb In a socalled 631G basis set in the GAUSSIAN program there are 6 Gaussians for core electrons and then 3 for Q8 and 1 for Qb The variation procedure applied to the HF wave functions The HF procedure uses the variational method to obtain the value of parameters that minimizes the energy SE 8lt lHl gt 8 IH d1 0 subject to the constraint the wave functions remain orthogonal lt il jgt I billde 5 The minimization of an equation subject to a constraint is carried out using the method of LaGrange undetermined multipliers 8 ltIgtIHltIgt gt EltltIgtIltIgt gt 1 0 Note that the multiplier is the energy E and it will be determined during the procedure The variational method in HF The variation can also be viewed as a variation of the function by an infinitesimal amount I ltlgt39 84v The energy then becomes Eltlgt 54gt ltltlgt 5 Hltlgt 5 gt ltltlgtHltlgtgt ltltlgt 5 Hlltlgtgt ltltlgtHltlgt 5 gt E SE In the variation method we are looking for the wave function I that will minimize the energy Here the condition is that the variation in the energy SE O as indicated on the previous slide This condition assures that E is stationary A stationary point is usually a minimum Introduction of the basis At this point we can substitute our trial wave function into the Fock equations E ltlIIllt gt g C CvltleHlxv gt The Lagrangian becomes L ltltlgtlHlltlgt gt E lt ltlgtlltlgt gt 1 g CZCVltxllHlxv gt 5 CZCvltxllxv gt 1 The first variation in the Lagrangian is set equal to zero 5L SCZCVltXMHXV gt CZ5CVltXMHXV gt 5C3CVltX Mlxv gt CZSCVltX Mlxv gt 0 There is an equation such as this for each possible linear combination and there are N possible linear combinations Expansion in terms of coefficients Since both E and L are real we can collect terms and exchange indices to obtain N 215CZH C ES C Kl23N p z Ky Hi 1 Ky pi HKp gt2 SW ltxKixM gt xixudr In other words H11 EZSHC11 H12 IiiSaca Hm EISWCM 0 H21 E1821Cu H22 IiiSum Hm EisgNCNi 0 Hm EZSMCU Hm EiSN2C2i HNN EZSNNCM 0 The secular determinant There are N equations and N 1 unknown variables C1iCziC3i CM and E In order for the equations to have meaningful nonzero solutions they must comprise a secular determinant ray 55H f Z ESU y ESE H th ESM E f ESM EQy ESB H u4da HQ Eamp1 Hg Eamp2 Hg Eamp3 H 0 which leads to N eigenvalues Ei i 123N The solutions for the bi C1ix1 02in CNixN under the constraint N N chcs1 u1v1 W W V Matrix representation The fock hamiltonian is an effective oneelectron hamiltonian He iZEi i He Efr AOS X19 X2 X39 quotquot9XN N Mos 4 Z CHIx1 1 The matrix representation for the overlap and interaction energies is Eigenvalues are energies and eigenvectors are MOs Thus HCi EiSCi Define the matrices of coefficients and eigenvalues as C11 C12 C13 C1N E1 0 0 O C 021 022 023 quot CZN E 0 E2 0 0 Cm CN2 CN3 CNN 0 0 0 EN Then the matrix form is HC SCE This system of equations is diagonalized if detH ES 0 However this is possible only in the MO basis A0 baSiS X17 X27 X37 39gtXN baSiS 117 127 137 gt N S W ltXMXVgt Not orthogonal 51 lt i jgt Orthogonal H W ltXHHe XVgt Not diagonal Eloy lt iHe Jgt Diagonal The Roothan matrix procedure The equations are solved on a computer using the Fock operator EV FHVCV 8 SWCV which is solved for all of the basis functions both occupied and unoccupied The coefficients obtained from this calculation are then used to calculate the Fock matrix whose elements are FM ltXHIhIXvgt ra xuxslCIIXVXKgt r iltxpxslglxlltxv YaK Z CSJCKJ Z C8iCKi g 1 r I occupied I 000 and same spin Ir r I Note only occupied basis functions in the Fock operator The Fock matrix is then substituted back into the first equation and coefficients are recalculated The procedure is carried out until a selfconsistent minimum energy is found The resulting selfconsistent field SCF energy is the HartreeFock procedure Summary of methods 1 Determine the optimum atomic orbitals This is done by a variational procedure for each atom The exponent of the STD or GTO is optimized More than one STO can be used per atomic orbital eg doublezeta basis The GTO requires parameterization of multiple Gaussian functions 2 Form linear combinations of the atomic orbitals at the positions indicated by a molecular geometry Note that this is initially just a guess and that the geometry which gives the lowest energy must be calculated in a number of tries cycles of the HF procedure 3 Perform the HF procedure repeatedly until a selfconsistent solution of the equations for the coefficients is obtained This is the selfconsistent field SCF method Nitrogen Molecular Orbitals Example of a Homonuclear Diatomic lsosurfaces represent W of orbital showing 90 of total probability The spatial wavefunction is an LCAO Core electrons are not included There are five electrons for each N atom Dinitrogen 10 MO N2 26 MO N217 MO This is a doubly degenerate orbital Only one of the two is shown N2 36 MO N2 27c MO 1 HHLLJHH I WWW TH W111 if I This is a doubly degenerate orbital Only one of the two is shown a Jam a in I ullhlnw lu h q l v lbw in m FEEEi gig lt 5 V an u mm I4olilll LIE u N2 46 MO Energy level diagram for N2 Negative energies represent bonding interactions lt 0 eV For N2 all there are ten electrons so all orbitals are filled through 3s Only valence orbitals are shown 46 175 l EU 36 17 26 lo Ground State Molecular Properties Bond length structure Vibrational frequency Calculated at stationary point Depends on accuracy of second derivative matrix with respect to nuclear displacement Dipole moment clearly zero for N2 Absorption spectrum Koopman s theorem We can think of the individual orbitals as representing the energy required to pull an electron off of the molecule out of an occupied orbital or add one to an unoccupied orbital Such an approach is a frozen electron approximation This means that we assume that none of the remaining electrons will respond to the electron removal or addition The formal statement of this approach is Koopman s theorem The ionization potential and electron affinity can be obtained From the energy of removal or addition of an electron to a molecular orbtial at the singledeterminant level HartreeFock is not well suited to calculation of excited state properties HartreeFock theory works well for ground state properties because the energies of occupied orbitals are relatively accurately determined However the energy of unoccupied orbitals is not well determined and therefore excited state properties and transition energies are not well determined within the HF approach To account for excited state properties one can include excited electron configurations Such ab initio approaches that move beyond HF theory are collectively called configuration interaction Cl Beyond HF Configuration Interaction The CI method employs a wavefunction which is constructed of a linear combination of the HF wavefunction with excited determinants C 000 vir occ vir b b PEC ZMWZCmW 1 l a The expansion coefficients are then selected so that they variationally minimize the expectation value of the electronic energy with respect to the Cl wavefunction C1 C1 C1 EGWMTgt where the wavefunction is restricted by the normalization condition vir C1 C1 2 occ vir 2 occ b 2 a a 1GWTgtQZ QZZCUW ljab Configuration Interaction for Singles If we imagine the case of H2 we can implement CI for singles by mixing in the excited state configurations ln molecules of high symmetry only the configurations of appropriate symmetry can mix in this way Cl that includes singles only is appropriate for improving transition energies but does i not help ground state properties This is known as Ground Brillouin s theorem State Mr 1 l lr tl Y Excited State Multielectron Configuration Interaction The CI expansion is variational and if the expansion is complete Full Cl gives the exact correlation energy within the basis set approximation The number of determinants in full Cl grows factorially with the system size making the method impractical for all but the smallest systems For this reason the Cl expansion is usually truncated at some order for example CISD where only singly and doubly excited determinants are considered Brillouin39s Theorem states that singly excited determinants do not mix with the HF determinant Therefore CISD is the cheapest worthwhile form of Cl yet this method scales as 0N6 where N is the size of the system
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