1、Basis Sets and Pseudopotentials,Slater-Type Orbitals (STOs),N is a normalization constanta, b, and c determine the angular momentum, i.e.L=a+b+c is the orbital exponent. It determines the size of the orbital.STO exhibits the correct short- and long-range behavior.Resembles H-like orbitals for 1sDiff
2、icult to integrate for polyatomics,Gaussian-Type Orbitals (GTOs),N is a normalization constanta, b, and c determine the angular momentum, i.e.L=a+b+c is the orbital exponent. It determines the size of the orbital.Smooth curve near r=0 instead of a cusp.Tail drops off faster a than Slater orbital.Eas
3、y to integrate.,Contracted Basis Sets,P=primitive, C=contractedReduces the number of basis functionsThe contraction coefficients, i, are constantCan be a segmented contraction or a general contraction,Contracted Basis Sets,Jensen, Figure 5.3, p. 202,STO-NG: STO approximated by linear combination of
4、N Gaussians,Even-tempered Basis Sets,Same functional form as the Gaussian functions used earlierThe exponent, , is fitted to two parameters with different and for s, p, d, etc. functions.Successive exponents are related by a geometric series- log() are evenly spaced,Reudenberg, K., et Al., Energy, S
5、tructure and Reactivity, Proceedings of the 1972 Boulder Conference; Wiley: New York, 1973. Reeves, C. M. J. Chem Phys. 1963, 39, 1.,Well-tempered Basis Sets, , , and are parameters optimized to minimize the SCFenergyExponents are shared for s, p, d, etc. functions,Huzinaga, S. et Al., Can. J. Chem.
6、 1985, 63, 1812.,Davidson, E. R.; Feller, D. Chem. Rev. 1986, 86, 681-696.,Used to model infinite systems (e.g. metals, crystals, etc.) In infinite systems, molecular orbitals become bands Electrons in bands can be described by a basis set of plane waves of the formThe wave vector k in a plane wave
7、function is similar to the orbital exponent in a Gaussian function Basis set size is related to the size of the unit cell rather than the number of atoms,Plane Wave Basis Sets,Polarization Functions,Similar exponent as valence function Higher angular momentum (l+1) Uncontracted Gaussian (coefficient
8、=1) Introduces flexibility in the wave functionby making it directional Important for modeling chemical bonds,Diffuse Functions,Smaller exponent than valence functions(larger spatial extent) Same angular momentum as valencefunctions Uncontracted Gaussian (coefficient=1) Useful for modeling anions, e
9、xcited states and weak (e.g., van der Waals) interactions,Cartesian vs. Spherical,Cartesians:s 1 functionp 3 functionsd 6 functionsf 10 functions,Sphericals:s 1 functionp 3 functionsd 5 functionsf 7 functions,Look at the d functions: In chemistry, there should be 5 d functions (usually chosen to be
10、, , , and . These are “pure angular momentum” functions.But it is easier to write a program to use Cartesian functions ( , , , , and .,Cartesian vs. Spherical,Suppose we calculated the energy of HCl using a cc-pVDZ basis set using Cartesians then again using sphericals. Which calculation produces th
11、e lower energy? Why?,Pople Basis Sets,Optimized using Hartree-Fock Names have the formk-nlm+G* or k-nlmG() k is the number of contracted Gaussians used for coreorbitals nl indicate a split valence nlm indicate a triple split valence + indicates diffuse functions on heavy atoms + indicates diffuse fu
12、nctions on heavy atoms and hydrogens,Pople Basis Sets,Examples:6-31G Three contracted Gaussians for the core with the valence represented by three contracted Gaussians and oneprimitive Gaussian 6-31G* Same basis set with a polarizing function added 6-31G(d) Same as 6-31G* 6-31G* Polarizing functions
13、 added to hydrogen and heavy atoms 6-31G(d,p) Same as 6-31G* 6-31+G 6-31G basis set with diffuse functions on hydrogen andheavy atoms The * notation is confusing and not used for larger basis sets:6-311+G(3df, 2pd),Dunning Correlatoin Consistent Basis Sets,Optimized using a correlated method (CIS, C
14、ISD, etc.) Names have the form aug-cc-pVnZ-dk “aug” denotes diffuse functions (optional) “cc” means “correlation consistent” “p” indicates polarization functions “VnZ” means “valence n zeta” where n is the number of functions used to describe a valence orbital “dk” indicates that the basis set was o
15、ptimized for relativistic calculations Very useful for correlated calculations, poor for HF Size of basis increases rapidly with n,Dunning Basis Sets,Examples:cc-pVDZ Double zeta with polarizationaug-cc-pVTZ Triple zeta with polarization anddiffuse functionscc-pV5Z-dk Quintuple zeta with polarizatio
16、n optimized for relativistic effects,Extrapolate to complete basis set limit,Most useful for electron correlation methodsP(lmax) = P(CBS) + A( lmax)-3 P(n) = P(CBS) + A( n)-3n refers to cc basis set level: for for DZ, 3 for TZ, etc. Best to use TZP and betterhttp:/molecularmodelingbasics.blogspot.dk
17、/2012/06/complete-basis-set-limit-extrapolation.htmlTCA, 99, 265 (1998),Basis Set Superposition Error,Occurs when a basis function centered at one nucleus contributes the the electron density around another nucleusArtificially lowers the total energyFrequently occurs when using an unnecessarily larg
18、e basis set (e.g. diffuse functions for a cation)Can be corrected for using the counterpoise correction.- Counterpoise usually overcorrects- Better to use a larger basis set,Counterpoise Correction,E(A)ab is the energy of fragment A with the basis functions for A+B E(A)a is the energy of fragment A
19、with the basis functions centered on fragment A E(B)ab and E(B)b are similarly defined,Additional Information,EMSL Basis Set Exchange:https:/bse.pnl.gov/bse/portalFurther reading:Davidson, E. R.; Feller, D. Chem. Rev. 1986, 86, 681-696.Jensen, F. “Introduction to Computational Chemistry”, 2nded., Wi
20、ley, 2009, Chapter 5.,Effective Core Potentials (ECPs) and Model Core Potentials (MCPs),Frozen Core Approximation,Approximation made: atomic core orbitals are not allowed to change upon molecular formation; all other orbitals stay orthogonal to these AOs,Pseudopotentials - ECPs,Effective core potent
21、ials (ECPs) are pseudopotentials that replace core electrons by a potential fit to all-electron calculations. Scalar relativisitc effects (e.g. mass-velocity and Darwin) are included via a fit to relativistic orbitals.Two schools of though: Shape consistent ECPs(e.g. LANLDZ RECP, etc.)Energy consist
22、ent ECPs(e.g. Stttgart LC/SC RECP, etc.),Shape Consistent ECPs,Nodeless pseudo-orbitals that resemble the valence orbitals in thebonding region,The fit is usually done to either the large component of the Dirac wavefunction or to a 3rd order Douglas-Kroll wave functionCreating a normalized shape con
23、sistent orbital requires mixing invirtual orbitalsUsually gives accurate bond lengths and structures,Energy Consistent ECPs,Approach that tries to reproduce the low-energy atomic spectrum(via correlated calculations),Usually fit to 3rd order Douglas-KrollDifference in correlation energy due to the n
24、odeless valence orbitals isincluded in the fitSmall cores are still sometimes necessary to obtain reliable results(e.g. actinides)Cheap core description allows for a good valence basis set (e.g. TZVP)Provides accurate results for many elements and bonding situations,Pseudo-orbitals,Visscher, L., “Re
25、lativisitic Electronic Structure Theory”, 2006 Winter School, Helkinki, Finland.,Large and Small Core ECPs,Jensen, Figure 5.7, p. 224.,Pseudopotentials - MCPs,Model Core Potentials (MCP) provide acomputationally feasible treatment of heavy elements.MCPs can be made to include scalar relativistic eff
26、ects- Mass-velocity terms- Darwin termsSpin orbit effects are neglected.- Inclusion of spin-orbit as a perturbation has beenproposedMCPs for elements up to and including the lanthanidesare as computationally demanding as large core ECPs.,MCP Formulation,All-electron (AE) Hamiltonian:,MCP Hamiltonian
27、:,First term is the 1 electron MCP HamiltonianSecond term is electron-electron repulsion (valence only)Third term is an effective nuclear repulsion,Huzinaga, S.; Klobukowski, M.; Sakai, Y. J. Phys. Chem. 1982, 88, 21. Mori, H; Eisaku, M Group Meeting, Nov. 8, 2006.,1-electron Hamiltonian,All-electro
28、n (AE) Hamiltonian:,MCP Hamiltonian:,First term is the 1 electron MCP HamiltonianSecond term is electron-electron repulsion (valence only)Third term is an effective nuclear repulsion,Huzinaga, S.; Klobukowski, M.; Sakai, Y. J. Phys. Chem. 1982, 88, 21. Mori, H; Eisaku, M Group Meeting, Nov. 8, 2006.
29、,MCP Nuclear Attraction,AI, I, BJ, and J are fitted MCP parametersMCP parameters are fitted to 3rd order Douglas-Kroll orbitals,Huzinaga, S.; Klobukowski, M.; Sakai, Y. J. Phys. Chem. 1982, 88, 21. Mori, H; Eisaku, M Group Meeting, Nov. 8, 2006.,MCP vs. ECP,ECPs “smooth out” the core, eliminating the radial nodal structureMCPs retain the correct radial nodal structure,Mori, H; Eisaku, M Group Meeting, Nov. 8, 2006.,
