Basis Set for Molecular Orbital Theory

Transcription

1 Basis Set for Molecular Orbital Theory! Different Types of Basis Functions! Different Types of Atom Center Basis Functions! Classifications of Gaussian Basis Sets! Pseudopotentials! Molecular Properties 1

2 Thermal Energy Enthalpy Free Energy Molecular Internal Energy System a collection of particles FREQ Quantum Particle atom or molecule OPT SCAN Ψ = Nuclear Degrees of Freedom E = Electronic Degrees of Freedom Total Energy of a Single Particle i c i ψ i ψ = φ N φ e i c i 2 E i E = E N + E e FREQ SP TD POLAR POP Electronic Energy (SCF, Single-point Energy) φ N = ϕ Vib ϕ Rot ϕ Tran E N = E Vib + E Rot + E Tran φ e = ϕ 1 ϕ 2 ϕ 3... E e = ε i 1 2 i V ij i< j AM1 PM3 HF DFT MP CI CC CBS Empirical Energy (Force Field) Translation (Particle in Box) FREQ Rotation (Rigid Rotor) FREQ Vibration (Harmonic Oscillator) FREQ Molecular Orbital (LCAO) STO-3G 3-21G 6-31G 6-31G(d,p) G(d,p) LANL2DZ UFF AMBER E(R), e.g. Lennard Jones Potential ϕ Tran = E Tran = h2 n 2 8ma 2 ϕ Rot = 2 nπ x sin a a 2l +1 4π (l m )! l (l + m l )! P m l (cosθ)e im l φ l E Rot = 2 2I l ( l +1 ) ϕ Vib = Atomic Orbital (Basis Set) 1 α 2 n n! π E Vib = hν(n ) ϕ i = c µ,i χ µ µ ε i = T i +V i + χ µ = j 1 4 V ij k H n (α 1 2 x)e α x2 2 a µ,k g k

3 Molecular Orbital Theory! Linear Combination of Basis Functions φ i M c µ, i χ µ µ =! Coefficients are variational parameters! ϕ i (MO) is initially unknown; describing the MO as a combination of known basis functions, χ! As M reaches the complete basis set limit, it is no longer an approximation. When M is finite, the representation is approximate. 3

4 Basis Functions M φi = µ c µ, iχ µ! {χ} are called the basis functions! Three criteria for selecting basis functions.! well-behaved! physically meaningful! computation of the integrals should be tractable 4

5 Types of Basis Functions! Plane Wave Basis Functions! Advantages:! Good for describing delocalized electron behavior! Free electron! Conjugation in polymers! Resonant states! Easy to apply periodic boundary condition good for systems with periodicity! Band structure! Band gap 5

6 Types of Basis Functions! Plane Wave Basis Functions! Disadvantages:! Bad for describing localized electronic behavior requires a linear combination of many basis functions! Core electron! Tight bonding! No analytical integrals done with numerical methods! Computational expensive! Numerical noise 6

7 Types of Basis Functions! Atom Centered Basis Functions! Advantages:! Good for describing localized electronic behavior! a few number of basis functions! most reactions at the molecular level! tight bonding! Have analytical integrals and derivatives for most methods! computational inexpensive! much less numerical noise 7

8 Types of Basis Functions! Atom Centered Basis Functions! Disadvantages:! Hard to describe hyper-delocalized electron behavior! Periodicity! Band structure! Bad gap! Free electron 8

9 Basis Set! Linear Combination of Atomic Orbitals (MO-LCAO) φ! Describing a MO as a linear combination of AOs {χ}.! minimal! double zeta / triple zeta / etc.! split valence i M c µ, i χ µ µ =! Special Basis Functions! polarization functions! diffuse functions 9

10 Basis Set Minimal Basis (Single Zeta)! Only enough functions are used to describe the electrons of the neutral atoms (core plus valence orbitals). Also known as single zeta basis set (zeta, ζ, is the exponent used in Slatertype orbitals)! H and He: 1s 1-AO! Li Ne: 1s, 2s, 2p 5-AOs! Na Ar: 1s, 2s, 2p, 3s, 3p 9-AOs! How about Fe? 10

11 Basis Set Multiple Zeta Basis (DZ, TZ, QZ )! Example HCN s p! Charge distributions are different in different parts of the molecule.! C-H σ-bond is made up of the H 1s orbital and the C 2p z. CN π-bond is made up of C and N 2p x and 2p y.! Because the π-bond is more diffusive, the optimal exponent for p x (p y ) should be smaller than that for the more localized p z orbital. 11

12 Basis Set Split Valence Basis! Only the valence part of the basis set is doubled.! Core orbitals are represented by a minimal basis.! VDZ double zeta 2x # of valence basis functions! H and He: 1s and 1s 2-AOs (same as DZ)! Li Ne: 1s, 2s and 2s, 2p and 2p 9-AOs! VTZ triplet zeta 3x # of valence basis functions! VQZ quadruple zeta 4x # of valence basis functions! V5Z quintuple zeta 5x # of valence basis functions! V6Z sextuple zeta 6x # of valence basis functions 12

13 Basis Set Multiple Zeta Basis (DZ, TZ, QZ )! DZ: Doubles the number of basis functions in a minima basis set! H and He: 1s and 1s 2-AOs! Li Ne: 1s and 1s, 2s and 2s, 2p and 2p 10-AOs! inner and outer functions the inner function has larger ζ exponent and is tighter (closer to the nucleus); the outer function has a smaller ζ, more diffuse (further from the nucleus)! Better description of the charge distribution compared to a minimal basis! TZ, QZ 13

14 Basis Set Special Functions! Polarization Functions additional functions with higher angular momentum added to a basis set to allow additional angular flexibility: molecules with strong local/global dipole moment! d orbitals for polarization of p orbital! p orbitals for polarization of s orbital! Diffuse Function additional functions with very small exponents added to a basis set to allow diffusive electron behavior: anions, dipole moment, polarizabilities! s orbitals with very small exponents for diffusion of s orbital! p orbitals with very small exponents for diffusion of p orbital 14

15 Contracted Basis Function! Each AO is made up of n GTO s of the same type n χ = cg i i= 1 i ζ ( ) 15

16 Gaussian Type Orbitals (GTO) ( ) = ( )( ) ς, nlm,, lm, ς, xyz,, l l x y lz ( ) = ( ) ( r R) ( r R) 2n 2 1 ς χ r, θ, ϕ NY θ, ϕ r R e Spherical Coordinate χ l l 2n 2 1 ς xyz,, Nx y z r R e Cartesian Coordinate 2 2! l x =l y =l z =0 s-orbital l x +l y +l z =1 p-orbital l x +l y +l z =2 d-orbital l x +l y +l z =3 f-orbital! Spherical d-function has five components (Y 2,2,Y 2,1,Y 2,0,Y 2,-1,Y 2,-2 ) Cartesian d-function has six components (x 2, y 2, z 2, xy, xz, yz)! Spherical f-function has 7 components Cartesian f-function has 10 components 16

17 Gaussian Type Orbitals (GTO)! Disadvantages:! Behavior near the nucleus is poorly represented.! GTOs diminish too rapidly with distance! Advantages:! GTOs have analytical integrals/derivatives! 17

18 Contracted Basis Function! Each AO is made up of n GTO s of the same type n χ = cg ζ ( ) i i i i= 1! If χ represents a p type AO, so are g PGTO s (primitive GTO)! all PGTO s in an AO have different exponents and coefficients! exponents and coefficients are constants defined in a basis set, obtained from atomic calculations, and adjusted for a representative set of molecules! contracted Gaussian functions! H and He: ls -> l's! Li Ne: mpns -> m'pn's! contraction scheme: [mpns/ls->m'pn's/l's] (Li-Ne/H-He) 18

19 Basis Set Nomenclature! Minimal Basis Set! STO-nG each Slater like AO is made up of n PGTO s! STO-3G (6s3p/3s)->[2s1p/1s]! General Multiple Zeta Basis! Usually defined with a contraction scheme! DZ (12s6p/6s)->[4s2p/2s]! TZ (24s12p/12s)->[6s3p/3s]! General Split Valance Basis! Usually defined with a contraction scheme! VDZ (9s6p/6s)->[3s2p/2s]! VTZ (16s12p/12s)->[4s3p/3s] 19

20 Basis Set Nomenclature! Where to look for basis set definitions?! 20

21 Basis Set Nomenclature! Pople Type Double Split Valence Basis k-nlg! 3-21G (6s3p/3s)->[3s2p/2s]! 3 PGTOs -> 1 core AO! 2 PGTOs -> 1 inner valence AO! 1 PGTO -> 1 outer valence AO! H and He: 1s(2) and 1s (1) 2-AOs 3-PGTOs! Li Ne: 1s(3),2s(2),2s (1),2p(2),2p (1) 9-AOs 15-PGTOs! Na Ar: 1s(3),2s(3),2p(3),3s(2),3s (1),3p(2),2p (1) 13-AOs 27-PGTOs! 6-31G (10s4p/4s)->[3s2p/2s]! 6 PGTOs -> 1 core AO! 3 PGTOs -> 1 inner valence AO! 1 PGTO -> 1 outer valence AO! H and He: 1s(3) and 1s (1) 2-AOs 4-PGTOs! Li Ne: 1s(6),2s(3),2s (1),2p(3),2p (1) 9-AOs 22-PGTOs! Na Ar: 1s(6),2s(6),2p(6),3s(3),3s (1),3p(3),2p (1) 13-AOs 46-PGTOs 21

22 Basis Set Nomenclature! Pople Type Triple Split Valence Basis k-nlmg! 6-311G (11s5p/4s)->[4s3p/3s]! 6 PGTOs -> 1 core AO! 3 PGTOs -> 1 inner valence AO! 1 PGTO -> 1 middle valence AO! 1 PGTO -> 1 outer valence AO! H and He: 1s(3),1s (1),1s"(1) 3-AOs 5-PGTOs! Li Ne: 1s(6),2s(3),2s (1),2s"(1),2p(3),2p (1),2p"(1) 13-AOs 26-PGTOs! Na Ar: 1s(6),2s(6),2p(6),3s(3),3s (1),2s"(1),3p(3),3p (1),3p"(1) 17-AOs 50-PGTOs 22

23 Basis Set Nomenclature! Pople Type Special Functions! Polarization Functions! * or (d) 1 set of d AO s for heavy atoms! ** or (d,p) 1 set of d AO s for heavy atoms; 1 set of p AO s for H and He! (2df,2pd) 2 sets of d and 1 set of f AO s for heavy atoms; 2 sets of p and 1 set of d AO s for H and He! Diffuse Functions! + 1 set of p and 1 set of s AO s with very small exponents for heavy atoms! ++ 1 set of p and 1 set of s AO s with very small exponents for heavy atoms; 1 set of s AO s with very small exponents for H (note: no diffuse function defined for He)! 6-31+G* or 6-31+G(d), 6-31+G** or 6-31+G(d,p), G(2df,2pd) 23

24 Basis Set Nomenclature! Correlation consistent (cc) basis sets (T.H. Dunning) recover the correlation energy of the valence electrons! cc-pvdz VDZ with 1 d- on heavy atoms;1 p- on H and He! cc-pvtz VTZ with 2 d- and 1 f- on heavy atoms; 2 p- and 1 d- on H and He! can be augmented with diffuse functions (aug-cc-pvxz)! aug-cc-pvdz additional 1 d-, 2 p-, and 2 s- with very small exponents on heavy atoms; 1 p- and 1 s- with very small exponents on H (note: elements He, Mg, Li, Be, and Na do not have diffuse functions defined within these basis sets.) Valence E CE ~65% ~85% ~93% ~95% ~98% 24

25 Basis Set Nomenclature! Other Important Basis Sets! DH Basis Sets! MINI, MIDI, and MAXI Basis Sets! Ahlrichs Basis Sets! ANO Basis Sets! Polarization Consistent Basis Sets 25

26 Basis Set Nomenclature Pseudopotentials! What is a pseudopotential?! A potential function form to replace the electrostatic potential of the nuclei and of the core electrons! Why do we need pseudopotentials?! To reduce the size of the basis set needed to represent the atom! To include relativistic effects for heavy elements! Why can we use pseudopotentials?! Core orbitals do not change much during chemical interactions! Valence orbitals feel the electrostatic potential of the nuclei and of the core electrons 26

27 Basis Set Nomenclature Pseudopotentials! Must use pseudopotentials for elements in the lower part of the periodic table! Some popular pseudopotentials or effective core potentials (ECP)! CEP-nG! LANL2DZ! Stuttgart 27

28 Basis Set Nomenclature Pseudopotentials! Zn core models! All electron model 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10! Small core model 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10! Large core model 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10! Au core models! 54-core-electron model [Xe]4f 14 5d 10 6s! 60-core-electron model [Kr]4d 10 5s 2 5p 6 4f 14 5d 10 6s! 78-core-electron model [Xe]4f 14 5d 10 6s 28

29 Molecular Properties Molecular Orbital φ i (r) = µ c µ,i χ µ (r)! iso-surface: a surface {r} where ϕ i (r) = constant! ϕ i (r) can have positive and negative values, shaded in different colors! only the change in sign (color) matters, not the absolute sign (color) 29

30 Molecular Properties Population Analysis! Mülliken Population Analysis (keyword POP) occ i occ i φφdτ= N * i i e { c χ }{ c χ } dτ = N * * µ, i µ ν, i ν e µ ν occ * { cµ, icν, i} { χµ χν dτ} = Pµν Sµν = ρµν = Ne µν i µν µν 1 1 ρ = ρ + ρ + ρ q A µν, µν, µν, µ A, ν A 2 µ A ν A 2 µ A ν A = Z ρ A A A! Natural Population Analysis (keyword POP=NBO or NPA) 30