One-Electron Hamiltonians

Size: px

Start display at page:

Download "One-Electron Hamiltonians"

Teresa Melton
5 years ago
Views:

1 One-Electron Hamiltonians Hartree-Fock and Density Func7onal Theory Christopher J MSSC, July 10, 2017

2 REVIEW A One-Slide Summary of Quantum Mechanics Fundamental Postulate: O Ψ = a Ψ What is Ψ? Ψ is an oracle! operator wave function (scalar) observable Where does Ψ come from? Ψ is refined Variational Process H Ψ = E Ψ 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 Construc7ng a 1-Electron Wave Func7on The units of the wave func;on are such that its square is electron per volume. As electrons are quantum par;cles with non-point distribu;ons, some;mes we say density or probability density instead of electron per volume (especially when there is more than one electron, since they are indis;nguishable as quantum par;cles) For instance, a valid trial wave func;on in cartesian coordinates for one electron might be: φ( x, y,z; Z) = 2Z 5 / 2 81 π % ' 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

4 Construc7ng a 1-Electron Wave Func7on A valid wave func;on in cartesian coordinates for one electron might be: φ( x, y,z; Z) = 2Z 5 / 2 81 π % ' 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 par;cular cartesian volume element (normaliza;on factors are determined by requiring that P = 1 when all limits are infinite, i.e., integra;on over all space)

5 Construc7ng a 1-Electron Wave Func7on To permit addi;onal flexibility, we may take our wave func;on to be a linear combina;on of some set of common basis func;ons, e.g., atomic orbitals (LCAO). Thus ( ) = a i ϕ( r) φ r N i=1 For example, consider the wave func;on for an electron in a C H bond. It could be represented by s and p func;ons on the atomic posi;ons, or s func;ons along the bond axis, or any other fashion convenient. C H

6 Construc7ng a 1-Electron Wave Func7on To op;mize the coefficients in our LCAO expansion, we use the varia;onal principle, which says that E = $ ' $ ' a i ϕ & i H & 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 H ϕ j dr ϕ i ϕ j dr = ij ij a i a j a i a j H ij S ij which is minimized by one root E of the secular equa;on (the other roots are excited states) each value of E that sa;sfies the secular equa;on determines all of its own a i values and thus the shape of the molecular orbital wave func;on (MO) H 11 ES 11 H 12 ES 12! H 1N ES 1N H 21 ES 21 H 22 ES 22! H 2N ES 2 N " " # " = 0 H N1 ES N1 H N2 ES N2! H NN ES NN

7 What Are These Integrals H? The electronic Hamiltonian includes kine;c energy, nuclear azrac;on, and, if there is more than one electron, electron-electron repulsion nuclei H ij = ϕ i 1 Z 2 2 ϕ j ϕ k i ϕ j + k r k " 2 electrons φ n ϕ i n r n ϕ j " The final term is problema;c. Solving for all electrons at once is a manybody problem that has not been solved even for classical par;cles. An approxima;on is to ignore the correlated mo;on 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 func;on expressed as an an;symmetrized product of one-electron MOs (a so-called Slater determinant).

8 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! 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 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 One MO per root E i = 0 a σi ( ) ( )

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

9 Cement Your Understanding F µν = µ 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)

10 Kinetic Energy Integral µ ν = 2Z A 6 Z A x x A 81 π ( ) 2 + ( y y A ) 2 + ( z z Z ) 2 ( y y A )e Z A ( x x A ) 2 + ( y y A ) 2 + ( z z Z ) x y z 2 ( ( ) 2 + ( y y ) 2 + z z B ( ) 2 ) B 2Z B 81 π 6 Z B x x B ( z z B )e Z B ( x x B ) 2 + ( y y B ) 2 + ( z z B ) 2 dx dydz µ on atom A, ν on atom B See why a digital computer is your friend? (There is actually an analytical solution to this integral!)

11 Electron Correla7on How Important is It? The fundamental approxima7on of the Hartree-Fock method: interac7ons between electrons are treated in an average way, not an instantaneous way f i = 1 2 i 2 Z k HF + V i j nuclei k r ik { } Infinite basis set results H One electron E HF = a.u. E exact = a.u. He Two electrons E HF = a.u. E exact = a.u. Error ~ 26 kcal mol 1!

12 Semiempirical MO (HF) Theory nuclei F µν = µ 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 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 Adopt a specific protocol (e.g., minimal basis) and parameterize matrix elements, or compute them from simple, analy7cal formulae, to facilitate Fock build and improve accuracy. E.g., AM1, PM3, PM6, OM2.

13 Fundamental Basis of Density Func7onal Theory Hohenberg and Kohn proved that the total energy can be determined exclusively from the electron density [ ] = T [ ρ( r) ] + V[ ρ( r) ] E ρ( r) conceptually simple but opera7onally challenging! In prac7ce one employs [ ] = T [ ρ * ( r) ] E ρ( r) ρ( r)z k dr + 1 r r k 2 nuclei k ρ( r)ρ r & r r & ( ) [ ( )] drd r & + V xc ρ r where ρ* is the correct density but for non-interac;ng electrons and V xc corrects for this approxima7on and for using the classical repulsion energy

14 Kohn-Sham SCF Kinetic energy of non-interacting electrons is simply sum of individual kinetic energies. To determine the KS orbitals use the same approach as in MO theory: Express KS orbitals within a set of basis functions {φ}, determine the individual orbital coefficients by solution of a secular equation entirely analogous to that employed in the HF theory. Replace the F µν elements by the the K µν elements K µν = φ µ nuclei k Z k r r k + ρ ( r' ) r r' dr'+v xc φ ν

15 Density Func7onal Theory No Panacea In its Kohn-Sham implementa7on, DFT also employs a oneelectron-operator formalism to compute densi7es/energies h i KS = 1 2 i 2 nuclei k Z k r k + ρ( r% ) r i r% dr% + V xc and the correct form for V xc is known only for very simple model systems (e.g., the uniform electron gas) PBE/cc-pV5Z H One electron E DFT = a.u. E exact = a.u. self-interac7on error! He Two electrons E DFT = a.u. E exact = a.u. Error ~ 7 kcal mol 1

16 HF vs DFT I Similarities between HF and KS Common variational principle (employed) Kinetic energy and nuclear attraction components of matrix elements of K are identical to those of F If the density in the classical interelectronic repulsion operator is expressed in the same basis functions used for the KS orbitals, then the same four-index electron-repulsion integrals will appear in K as in F (much faster if auxiliary basis is used for density) Density required for computation of the secular matrix elements Density determined using the orbitals obtained from the solution of the secular equation KS procedure is a SCF iterative procedure Historically: modify existing codes for HF calculations to perform DFT calculations

17 HF vs DFT II Key differences between HF and KS DFT as derived so far contains no approximations: it is exact. But, we need to know E xc as a function of ρ. Hohenberg and Kohn proved that a functional of ρ must exist. No guidance, though, as to what the functional should look like. Contrast between HF and DFT: HF approximate theory: solve the relevant equations exactly. DFT exact theory: solve the relevant equations approximately because the form of the operator is unknown. Exact DFT is variational. When approximations of E xc are introduced, this is no longer true. Both exact and approximate DFT are size extensive.

18 Simplest functional: LSDA local spin-density approximation Kohn & Sham 1965, von Barth & Hedin 1972 Example: for a closed-shell system: Energy = T KS +V en +V ee + d 3! r ρ r! ε x ρ r! density ( ){ [ ( )] +ε c [ ρ ( r! )]} Exchange energy of uniform electron gas neutralized by uniform positive background charge = ρ 1/3 Correlation energy. Fit to Monte Carlo calculations for uniform electron gas

19 Density Gradient Correc7ons In a molecule the electron density is not spatially uniform. LDA has serious limitations for energies, although it gives good geometries. Improve functionals by making them depend on the extent to which the density is locally changing, i.e. the gradient of the density. Functionals that depend on both the density and the gradient of the density: gradient corrected or generalized gradient approximation (GGA) functionals. Most GGA functionals are constructed with the correction being a term added to the LDA functional ε x /c [ ( )] = ε x /c GGA ρ r LSD ρ( r) [ ρ( r) ] + Δε x /c ρ 4 3 ( r) x/c: same functional for either exchange or correlation. The dependence of the correction term is on the dimensionless reduced gradient. ( 28) verify that this term is dimensionless

20 Generalized gradient approximation shows promise Mean (unsigned) errors in kcal/mol Bond Barrier energies heights Hartree-Fock theory 31 9 Local spin-density approximation Correct thru 2nd order: SOGGA 7 13 GGA: BLYP (1988) Becke-Lee-Yang-Parr This aroused the attention of many quantum chemists.

21 Density Gradient Correc7ons II Taylor-function-expansion justification for the importance of the gradient. Obvious next step: include second derivative of the density, i.e. the Laplacian. Meta-GGA because they go beyond the gradient correction. Alternative Meta-GGA formalism, numerically more stable is to include in the exchange-correlation potential a dependence on the kinetic-energy density τ: τ ( r) = occ 1 ψ i 2 Functions ψ are the self-consistently determined KS orbitals. i ( r) 2 Some examples of MGGA functionals for exchange, correlation, or both, are B95, B98, ISM, τhcth, and the Minnesota local functionals M06-L and M11-L. Cost of MGGA comparable to that for GGA. MGGA generally more accurate than GGA, but numerically more demanding.

22 Adiaba7c Connec7on B3LYP model (one of first hybrids ) is defined as E B3LYP xc = ( 1 a)e LSDA x + ae HF x + bδe B x + ( 1 c)e LSDA LYP c + ce c ( 35) a,b,c same values as in B3PW91. Of all the modern functionals B3LYP has proven the most popular to date, although its reign seems to be ending (but B3LYP-D3 bringing second life). Adiabatic Connection methods incorporate both HF and DFT: hybrid methods (including HF means N 4 scaling, so less frequently used in solid-state calcs).

23 Older popular functionals have deficiencies: #1 No functional prior to 2005 is good for both barrier heights and transition metal bond energies. The best functionals for organic chemistry are bad for transition metal bond energies and vice versa. #2 Most functionals prior to 2005 are bad for barrier heights and noncovalent interactions, especially π-π stacking. #3 Standard functionals are terrible for charge-transfer excitations. #4 (a practical issue) DFT with Hartree-Fock exchange is too expensive for many applications.

24 Amino acid residue pair with both electrostatics and π π stacking 6 best estimate: 5.3 Molecular-mechanics-like addon dispersion terms (protocols) have been developed for older functionals (-D, -D2, -D3) binding energy (kcal/mol) tyrosine phenylalanine BLYP B3LYP PBE B98 PBEh TPSSh BMK M05-2X M06-L M06-HF M06 M06-2X -3

25 THANKS!! Onwards

A One-Slide Summary of Quantum Mechanics

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!