A One-Slide Summary of Quantum Mechanics

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2 A One-Slide Summary of Quantum Mechanics Fundamental Postulate: O! = a! What is!?! is an oracle! operator wave function (scalar) observable Where does! come from?! is refined Variational Process H! = E! electronic road map: systematically improvable by going to higher resolution convergence of E Energy (cannot go lower than "true" energy) Hamiltonian operator (systematically improvable) truth What if I can't converge E? Test your oracle with a question to which you already know the right answer...

3 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 ) 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)

4 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 #( r) " r N $ i=1 For example, consider the wave function for an electron in a C H 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 H

5 What Are These Integrals H? The electronic Hamiltonian includes kinetic energy, nuclear attraction, and, if there is more than one electron, electron-electron repulsion H ij = " i # 1 nuclei electrons $ 2 Z " j # " k & 2 % i " j + " % i " j 2 k r k n r n 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 Hartree-Fock approach accomplishes this for a many-electron wave function expressed as an antisymmetrized product of one-electron MOs (a so-called Slater determinant)

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11 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 Hartree- Fock procedure Construct and solve Hartree- 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 L F 1N ES 1N F 21 ES 21 F 22 ES 22 L F 2N ES 2N M M O M F N1 ES N1 F N2 ES N2 L F NN ES NN nuclei F µ! = µ 1 2 "2! # Z k µ 1! r k ( µ! "#) = %% $ µ ( 1)$! 1 + P & # $% µ! $% $% ' ( ) 1 k ( ) 1 2 ( µ$!%) occupied P!" = 2 # a!i a "i $ " ( 2)$ # ( 2)dr ( 1)dr 2 r 12 i = 0 ( ) One MO per root E ( )

12 Wavefunction Basis set set of mathematical functions from which the wavefunction is constructed Linear Combination of Atomic Orbitals (LCAO)

13 What functions to use? Slater Type Orbitals (STO) % " 1s = ' # & $ Can t do 2 electron integrals analytically %% $ 1 r ( * ) 1 2 e # +r ( 1) $ ( 1) ( 2) ( 2) # $ "! dr1 dr2 µ $ 12

14 Fock Operator Treat electrons as average field F µ" = µ # $2 2 " #& µ % k r " + P ) & '( µ" '( # 1 * + 2 k '(, µ' "( -. Density Matrix occupied MOs $ P "# = 2 a "i a #i i 2 electron integral 2 electron integrals scale as N 4

15 1950s Replace with something similar that is analytical: a gaussian function % " 1s = ' # & $ ( * ) 1 2 e # +r VS. % " "1s" = ' 2# & $ ( * ) 3 4 e +#r 2

16 Contracted Basis Set STO-#G - minimal basis Pople - optimized a and α values N & " = # a 2$ i ) "1s" i ( + ' % * i 3 4 e,$ ir 2

17 H F vs. H H What do you about very different bonding situations? Have more than one 1s orbital Multiple-ζ(zeta) basis set Multiple functions for the same atomic orbital

18 Double-ζ one loose, one tight Adds flexibility Triple-ζ one loose, one medium, one tight Only for valence

19 Decontraction Allow a i to vary locked in STO-3G STO#3G " 1sH = 3 ' $ 2% ai i ) ( & i *, e #% ir 2 primitive gaussians Pople - #-##G 3-21G gaussian valence: 2 tight, 1 loose primitives in all core functions

20 How Many Basis Functions for NH 3 using 3-21G? NH G atom # atoms AO degeneracy basis fxns primitives total basis fxns total primitives N 1 1s(core) s(val) = p(val) = H 3 1s(val) = total = 15 24

21 Polarization Functions 1 d functions on all heavy atoms (6-fold deg.) 6-31G** 1 p functions on all H (3-fold deg.) 6 primitives 1 core basis functions 2 valence basis functions one with 3 primitives, the other with 1

22 For HF, NH 3 is planar with infinite basis set of s and p basis functions!!!!! O + H O H = O Better way to write 6-31G(3d2f, 2p) Keep balanced Valence split polarization 2 d, p 3 2df, 2pd 4 3d2fg, 3p2df

23 Dunning Basis Sets cc-pvnz N = D, T, Q, 5, 6 polarized correlation consistent

24 loose electrons anions excited states Rydberg states Diffuse Functions Dunning - aug-cc-pvnz Augmented Pople 6-31+G - heavy atoms/only with valence G - hydrogens Not too useful

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