Solving Hartree Fock Equation using a Basis Set HF Roothaan equation 1 HF Roothaan Equation For molecules numerical basis not efficient so use atom centered basis set to describe the molecular orbital N b a s i s a r1 C ua u a 1, 2 ,...N b a s is u 1 f r1 a r1 aa r1 Nbasis Nbasis f r1 Cua u a Cua u u 1 u 1 Nbasis Nbasis C * r f r r dr C * r r dr ua v 1 1 u 1 1 a ua v 1 u 1 1 u 1 u 1 Nbasis Nbasis CuaFvu a CuaSvu a 1,2,...Nbasis u 1 u 1 푭푪 = 푺푪흐 2 Self Consistent Field We just have to solve the fock equation: PROBLEM FOCK OPERATOR HAS THE SOLUTION INSIDE F C C SC So put in a guess Cguess this allows you to get C1 F C g u e s s C 1 SC 1 Then put in C1 this allows you to get C2 Continue the cycle until you get convergence on Cinput and Coutput SELF CONSISTENT FIELD (SCF) method 3 Fock Operator in Basis Representation 푎, 푏 are for spin orbitals 푢, 푣 are for basis sets Core Hamiltonian Matrix Two electron Matrix n / 2 core * r 2J r K r r dr huv v * r1 hr1 u r1 dr v 1 b 1 b 1 u 1 1 b1 N n / 2 1 2 Z I hr1 1 2 vb| ub vb| bu 2 I 1 r 1I b1 Calculated once in a n / 2 Nbasis Nbasis C wb *C xb 2 vw| ux vw| xu SCF cycle b1 w1 x1 4 Fock Operator in Basis Representation F * r f r r dr * r h r r dr vu v 1 1 u 1 v 1 1 u 1 n / 2 Nbasis Nbasis Cwb *Cxb 2 vw| ux vw| xu b1 w1 x1 Nbasis Nbasis hvu Pwx 2 vw| ux vw| xu w1 x1 Where we define the density matrix as n / 2 Pwx Cwb *C xb b1 Since C changes every iteration this has to be recalculated every time 5 Self Consistent Field Method vw| ux * r * r r 1 r r dr dr v 1 w 2 12 u 1 x 2 1 2 Calculate initially S d r * F vu v * r 1 f r 1 u r 1 d r 1 vu 1 v u F C C SC UPDATE Get newest results 6 Basis Sets 7 Slater Type Orbital vs Gaussian Type Orbital In the H2 case we used Slater type orbitals for radial part where basis on HA is given as 1 / 2 STO 3 r R A 1 S , r R A / e Another tye is a Gaussian type orbital for the radial part where basis on H Ais given as 2 GTO 3 / 4 r R A 1 S , r R A 2 / e 8 GTO vs STO STO GTO STO larger slope near r=0 d STO , r R d G TO , r R 1S A 0 1S A 0 dr r 0 dr r 0 9 Solve the hydrogen atom by Gaussian Function 2 e2 2ll 1 Hˆ r 2 2 2 2r r r 40r 2r 1S l=0, using atomic units 1 2 1 Hˆ Hˆ r trial trial 2 E 2r r r r trial trial trial 2 using trial Expr Solve for variational minimum of and obtain the minimum energy. Compare the value with the exact one. 10 Hint: Gaussian Integrals Hˆ exp r 2 Hˆ exp r 2 r 2 dr trial trial 0 exp r 2 exp r 2 r 2 dr trial trial 0 r n r n exp 2r 2 dr 0 1 r1 4 1 r 2 8 2 4 3 r 2 32 2 11 Why GTO? 4 orbital basis integral: if the basis are on 4 different atoms vw| ux * r * r r 1 r r dr dr v 1 w 2 12 u 1 x 2 1 2 GTO GTO GTO 1 S , r R A 1 S , r R B K AB 1 S , r R D R A R 3 / 4 2 R D ; K AB 2 / exp R A R B 1.4 1.2 GTO1 GTO2 1 GTO1GTO2 0.8 0.6 0.4 0.2 0 -3 -2 -1 0 1 2 3 12 Linear Combination of GTO Contracted GTO: One GTO is not enough to describe the STO type orbital so lets use more than one GTO and add with correct coefficient (d) and exponent (a) value L CGTO GTO r R A d p p , r R A p 1 13 Basis Sets • Minimal Basis: only those in atomic orbital so one 1S orbital for hydrogen, one 1S, 2S, 2P for carbon, oxygen • STO-3G • Split Valence: valence orbital has two, for hydrogen two 1S orbital, one 1S and two 2S, 2P for carbon oxygen • 6-31G, 3-21G • Diffuse: large version of valence orbital • +, ++ • Polarization: higher angular momentum add 2P for hydrogen, add 3D for carbon, oxygen • *,**, (d), (d,p) 14 Dunning Correlation Consistent Basis Set T. Dunning decided on defining the contraction coefficient and exponential coefficient to maximize electron correlation aug-cc-pVDZ (X=2) aug-cc-pVTZ (X=4) aug-cc-pVQZ (X=4) aug-cc-pV5Z (X=5) 2 E X E CBS B e xp X 1 C e xp X 1 15 Basis Set Effective Core Potential • Most Chemistry occurs between valance electrons, electrons in the core do not contribute to important reactions use EFFECTIVE POTENTIAL FOR THE CORE ELECTRONS: Effective Core Potential (ECP) used for many electron systems. Relativistic Effect become important for heavy transition metal systems so use relativistic ECP (RECP) Hay-Wadt: LANL2DZ LANL2TZ, Dolg In bulk simulation like VASP it is called Psuedo potential 16 Basis Set Library (https://bse.pnl.gov/bse/portal) 17 Pick Basis Set Convergence Dipole moment of H2O Method # of basis Debye HF/STO-3G 7 1.7076 HF/6-31G 13 2.5006 HF/6-31+G 17 2.5824 HF/6-31+G(d,p) 29 2.2339 HF/6-311G 19 2.4881 HF/6-311++G 25 2.5536 HF/6-311++G(d,p) 37 2.1959 HF/6-311++G(3d,3p) 61 1.9744 HF/6-311++G(3df,3pd) 83 1.9681 HF/aug-cc-pVTZ 105 1.9394 HF/aug-cc-pVQZ 215 1.9361 Exp 1.8550 Computational time ~ (# of basis)2 18 1 1 1 2 1 Hˆ r 2 2r r 2 2 2 2 2r r r r 2r r r r 1 1 2 1 2 r r 2 r r 1 1 2 1 Hˆ exp r 2 exp r 2 r 2dr trial trial 2 r r 2 r r expr 2 2r expr 2 r 2 expr 2 2 4 2r 2 expr 2 r 2 ˆ 2 2 2 1 2 2 trial H trial expr 2 2 r expr r dr r 3 r 2 2 2 r 4 r1 19 3 3 1 Hˆ 3 r 2 2 2 r 4 r1 trial trial 8 2 16 2 4 3 1 16 2 4 3 1 Hˆ trial trial 16 2 4 2 1 trial trial r 1 8 2 trial trial 8 2 3 2 r n r n exp 2r 2 dr 2 2 0 Derivative with respect to 1 r1 dE 3 2 1 2 2 4 0 d 2 3 2 1 r 8 24 16 8 2E 2 9 18 9 2 4 3 r 2 32 2 (ans -0.424, 0.076 hartree off from the true value of -0.5) 20.