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Running a calculation,For ab initio or DFT,calculations many. programs require a basis set,choice to be made,The basis set is an approx. imate representation of the,atomic orbitals AOs,The program then calculates. molecular orbitals MOs,using the Linear Combin,ation of Atomic Orbitals. LCAO approximation,CCCE 2008 4,Computational Chemistry Map.

Chemist Decides Computer calculates,Starting Molecular. AOs determine the,Geometry O wavefunction,Basis Set with H H. ab initio and DFT,Type of Calculation LCAO,Method and. Assumptions O,Properties to H H,be Calculated MOs,CCCE 2008 5. Critical Choices,Choice of the method and basis set used is.

Which method,Molecular Mechanics Ab initio Semiempirical or. Which approximation,MM2 MM3 HF AM1 PM3 or B3LYP etc. Which basis set if applicable,Minimal basis set,Split valence. Polarized Diffuse High Angular Momentum,CCCE 2008 6. Why is Basis Set Choice Critical,The basis set needs to be able to approximate.

the actual wave function sufficiently well to,give chemically meaningful results. Also needs a reasonable computational cost,Integrals should be evaluated quickly and. accurately,Trade offs,Choice always involves a balance between. accuracy and computational cost, More accurate methods using larger basis sets will. take more computer time,CCCE 2008 7,Theoretical Models.

Goal of computational chemistry is to,mathematically represent chemical reality. Improving the basis set and the degree of electron. correlation improves the ability of the,computational model to approach reality. Ultimate goal is an exact solution of the,Schr dinger equation. CCCE 2008 8,Comparison of Some Methods for Accuracy. DFT Location,HF MP2 MP3 MP4 QCISD T Full CI,Minimal Electron Correlation.

A Split Valence,S 6 311G d p G,T 6 311 G d p L,6 311 G 2d p. 6 311 G 3df 3dp,HF Schr dinger,Limit Equation,CCCE 2008 9. Possible Basis Functions,1 Hydrogen like Orbitals,Derived for a one electron atom. Not truly accurate for a many electron atom,Form r R r Yl m. R r radial function,Yl m spherical harmonic,Advantages Mutually orthogonal.

Disadvantages Complex form is awkward for,calculations Most atoms of interest one. CCCE 2008 10,Possible Basis Functions,True wavefunction should be antisymmetric to. electron interchange use spin orbitals,Antisymmetric linear combination of products of. spin orbitals used in an SCF calculation,HF SCF calculation. Numerical methods were originally used to solve and. find the Hartree Fock orbitals,Roothaan Represent the HF orbitals as linear.

combinations of a set of known basis functions, Commonly used set of basis functions for atomic HF. calculations is the set of Slater type orbitals STOs. CCCE 2008 11,Possible Basis Functions,2 Slater Type Orbitals STOs. Normalized form,2 a0 n 0 5 n 1 r a0 m,where n m and l are integers and orbital. exponent is a variational parameter,Improve results by using a linear combination of. several STOs to represent each HF orbital,HF SCF atomic calculations require lots of.

computation,Hartree did this numerically in the 1930 s. CCCE 2008 12,Slater Type Orbitals,Advantages Have a complete set. Radial behavior closely matches hydrogenic,Disadvantages. No nodes as with H like orbitals,Not mutually orthogonal. For larger molecules computer evaluation of the,many integrals involved is quite time consuming.

Need to reduce the computational cost,CCCE 2008 13. Possible Basis Functions,3 Gaussian Type Orbitals GTOs. Proposed by S F Boys in 1950,2 2 n 1 5 2 n 1 4 n 1 r 2 m. g r r e Yl,Advantages Have a complete set,Computer evaluation of integrals much faster. Closed integrals Integrated GTO gives a GTO,Disadvantages Not mutually orthogonal.

Representation of e probability is poor near and,far away from the nucleus. CCCE 2008 14,Comparison,STO vs GTO,0 8 GTO e r2 STO. 0 0 5 1 1 5 2 2 5 3 3 5 4 4 5,CCCE 2008 15,Linear combinations of GTO s are used to. approximate STOs which are themselves,approximations. A single GTO basis function has significant errors. when compared to a STO especially near the,nucleus See previous slide.

If several GTOs are combined in a linear,combination the basis function is greatly. See next slide,CCCE 2008 16,Comparisons,STO vs GTOs. Wavefunction,0 1 2 3 4 5 6 7,Radius Angstroms,CCCE 2008 17. Use of GTOs,Individual GTOs not used as basis functions. Use a normalized linear combination of a few,GTOs called primitives each with different.

values to give a contracted Gaussian function,gc c g i p. where gc is a contracted gaussian,g p is a primitive gaussian and ci is. a contraction coefficient, A linear combination of these primitives typically. 1 7 is used to approximate the STO,CCCE 2008 18,Use of GTOs. Using contracted GTOs instead of primitive,GTOs as the basis set has advantages.

Number of variational coefficients to be,determined is reduced which saves a lot of. computational time,Accuracy is NOT reduced significantly as long as. the contraction coefficients ci s are well chosen,Increasing the number of primitive GTOs used in. each contracted Gaussian improves the accuracy, Different types of basis sets use different numbers. and types of GTOs,CCCE 2008 19,Minimal Basis Sets,Jargon Used STO NG Single.

N is the number of primitive GTOs used,Example STO 3G. Three primitive GTOs used per AO,Popular starting point for calculations. STO 3G basis functions have been developed for,most of the elements in the Periodic table. Minimal basis sets do not adequately describe, non spherical anisotropic electron distribution in. molecules as in polar covalent bonds,CCCE 2008 20,Minimal Basis Sets.

GTO representation of a STO for 1s AO,r 0 5 1 5e r where 1. 2 4 2 2 2 4 2 4 2,1sSTO 3G r c1 1 e 1r c2 2 e 2r c3 3 e 3r. where c1 0 444615 c2 0 535336 c3 0154340,and 1 0109818. 2 0 405771 3 0 2 22766,c values are called the contraction coefficients. The exponents are the alpha values,CCCE 2008 21,Split Valence Basis Sets.

Jargon Used K LMG Double,Differentiate between core and valence electrons. Developed to overcome problems of inadequate,description of anisotropic electron distributions. using minimal basis sets Size is adjusted,K number of sp type inner shell primitive GTOs. L number of inner valence s and p type primitive,M number of outer valence s and p type. primitive GTOs,CCCE 2008 22,Split Valence Basis Sets.

Each split valence atomic orbital is composed of a. variable proportion of two or more functions of,different size or radial extent. For a larger e cloud,longer bond,For a smaller e cloud. shorter bond,a and b are normalized and sum to 1,CCCE 2008 23. Split Valence Basis Sets,3 21G Used as the semiempirical basis set. Three primitives for the inner shell STO 3G each,valence orbital is constructed with two sizes of.

basis function Two GTOs for contracted valence,orbitals One GTO for extended valence orbitals. STO 6G for inner shell Three sizes of basis,function for each valence orbital Three GTOs for. contracted valence orbitals and two different sizes. of GTO for extended valence orbitals,CCCE 2008 24,Polarized Basis Sets. Jargon Used 6 31G d or 6 31G older,Also have 6 31G d p or 6 31G. d type functions added to atoms with Z 2,f type functions added to transition metals.

d p or type,p type functions added to H atoms,d type functions added to atoms with Z 2. f type functions added to transition metals,6 31G d is another popular basis set choice. CCCE 2008 25,Polarized Basis Sets,In molecule formation AOs become distorted. in shape polarization,Orbitals are influenced by other nuclei. Polarization accounts for these influences which,distort the orbital shape.

CCCE 2008 26,Diffuse Basis Sets,Jargon Used 6 31 G d or 6 31 G d. 6 31G d basis set with an additional larger p,function for atoms with Z 2. 6 31 G d basis set with an additional larger s,function for H atoms. Diffuse basis sets are useful for describing,anions molecules with lone pairs excited. states and transition states loosely held e,CCCE 2008 27.

Basis Set Progression,Increasing number of GTOs Used. Minimal Split Valence Polarized Diffuse, Get an increasingly good approximation to the actual. wave function,The number of integrals increases as N4 where N. is the number of basis functions,During the minimization process the orbital. exponents are adjusted to define a new basis set to. start another iteration,Computational cost has to be considered.

CCCE 2008 28,Comparison of Some Methods for Accuracy. DFT Location,HF MP2 MP3 MP4 QCISD T Full CI,STO 3G Electron Correlation. A Split Valence,I Polarized,S 6 31G d DZVP,6 311G d p TZVP G. E Diffuse O,T 6 311 G d p A,6 311 G 2d p,6 311 G 3df 3dp. HF Schr dinger,Limit Equation,CCCE 2008 29,Basis Set Choice and Expense.

axial methylcyclohexane on SGI Indigo2,Spartan cpu time in sec. Method Basis Set s p opt,AM1 STO 3G 1 10,HF STO 3G 72 983. HF 3 21G d 193 2214,HF 6 31G d p 2632 34655 9 6 h, As larger basis sets are used the energy decreases. Variational Principle,CCCE 2008 30,Common Basis Sets. Brief Description of Standard Basis Sets,Basis Set Description.

STO 3G Minimal basis qualitative results large systems. 3 21G Double more accurate results on large systems. 6 31G d Moderate set Common use for medium systems. 6 31G d p Used where H is site of interest More accurate. 6 31 G d Used with anions excited states lone pairs etc. 6 31 G d p Used with anions etc where H is site of interest. 6 311 G d p Good for final accurate energies but expensive. Many other sets are in use Existing sets can be modified. CCCE 2008 31,Optimization of a Basis Set,Variational Principle Energy values are. bounded from below,The lower the calculated energy the better. Vary the constants and exponents that describe the. Gaussian functions sequentially until the lowest,energy is obtained. Such a basis set may only apply to that,individual molecule however. CCCE 2008 32,Lab Gaussian Orbitals,Question How are Gaussian orbitals used to.

approximate a Slater Type Orbital,Importance Basis sets are approximations. based on mathematical use of two or more,Gaussian functions. Goal Visualize an STO 3G basis set,what does the resultant function look like. Computational Tool Spreadsheet,Refer to Basis Set Case Study handout for. Basis Sets CCCE 2008 2 Session 2 Basis Sets Two of the major methods ab initio and DFT require some understanding of basis sets and basis functions This session describes the essentials of basis sets What they are How they are constructed How they are used Significance in choice CCCE 2008 3 Running a Calculation In performing ab initio and DFT computa tional

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