量子化学简介1113

15
Split Basis Sets...

3-21G
–பைடு நூலகம்
– – –
16
commonly used simple split basis set; OK for HF geometry calculations on 1st row elements, not good for heavier elements or for accurate energies 3 primitive Gaussian functions for inner core (subvalence) electrons 2 Gaussians for contracted (small) valence orbitals 1 Gaussian for extended (large) valence orbitals

Assumes that each electron experiences all the others only as a whole (field of charge) rather than individual electron-electron interactions. Introduces a Fock operator F: F which is the sum of the kinetic energy of an electron, a potential that one electron would experience for a fixed nucleus, and an average of the effects of the other electrons.
Kinetic energy Potential energy
(Tn is omitted; this ignores relativistic effects, yielding the electronic Schrödinger equation.)
8
Hartree-Fock Approximation
量子化学简介
1
量子化学概述
量子化学是用量子力学原理和方法研究分子微观结构 以及与物质的物理和化学性质关系的一门边缘学科。 在它过去多年的发展过程中，建立了一系列理论和计 算方法，概括出许多对化学发展有重要意义的概念和 规律。特别是七十年代以来，随着计算机技术的发展， 量子化学已进入一个新的发展阶段，其研究成果使人 们对物质结构与性质的内部联系以及化学现象本质与 规律性的认识大大深化了。

electrons
1 rab
Basis of M.O. Theory...

Simplifying assumptions are employed to ‘solve’ the Schrödinger equation approximately: – Born-Oppenheimer approximation allows separate treatment of nuclei and electrons – Hartree-Fock independent electron approximation allows each electron to be considered as being affected by the sum (field) of all other electrons. – LCAO Approximation
2
理论方法

从头算分子轨道理论（Ab Initio Molecular Orbital Theory） 半经验的分子轨道理论（SemiEmpirical Molecular Orbital Theory）


密度泛函理论（Density Functional Theory)
3
分子轨道理论

Schrödinger equation:
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Split Basis Sets


Minimal (small) basis sets such as STO-3G do not adequately describe non-spherical (anisotropic) electron distribution in molecules „Split‟ valence basis sets (3-21G; 6-31G, etc.) were developed to overcome this problem Each split valence atomic orbital is composed of a variable proportion of two (or more) functions of different size or radial extent
7
Born-Oppenheimer Approx.


States that electron motion is independent of nuclear motion, thus the energies of the two are uncoupled and can be calculated separately. Derives from the large difference in the mass of nuclei and electrons, and the assumption that the motion of nuclei can be ignored because they move very slowly compared to electrons Htot a (Tn) + Te + Vne + Vn + Ve
Schrödinger equation after BornOppenheimer Approximation
0
_ h2 H = 82
nuclei
kinetic energy (nuc.)

A
1 MA
2
A
kinetic
ZAZB rAB
_ h2 82m
electrons
a energy

nuclei electrons

9
LCAO Approximation

Electron positions in molecular orbitals can be approximated by a Linear Combination of Atomic Orbitals. This reduces the problem of finding the best functional form for the molecular orbitals to the much simpler one of optimizing a set of coefficients (cn) in a linear equation:
Gaussian type orbitals (GTO)

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Gaussian Type Orbitals

STO-3G
– –
–
Slater type orbitals approximated by three primitive Gaussian functions Used as default basis set in semi-empirical MO calculations (AM1, PM3) Better approximations using combinations of Gaussian functions have been developed and are generally employed in ab initio work
12
Basis Sets

Combinations of mathematical functions used to represent atomic orbitals
–
Minimal

H: 1s C, N, O: 1s, 2s, 2px, 2py, 2pz (all three needed to maintain spherical symmetry)
2 a
_ e2
(elect.)

A a
ZA rAa
+ e2

nuclei
+ e2
constant
A > B
6
a > b 1 kinetic energy term plus 2 Coulombic energy terms: (one attractive, 1 repulsive) plus a constant for nuclei
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Hartree-Fock Self-Consistent Field (SCF) Method...

Computational methodology:
– – – –
guess the orbital occupation (position) of an electron guess the potential each electron would experience from all other electrons (taken as a group) solve for Fock operators to generate a new, improved guess at the positions of the electrons repeat above two steps until the wavefunction for the electrons is consistent with the field that it and the other electrons produce (SCF).