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1、第五章 关联,5.1 单电子近似的理论基础 5.2 费米液体理论 5.3 强关联体系,多电子体系(After Born-Oppenheimer 绝热近似):,5.1 单电子近似的理论基础,关联：电子电子相互作用,弱：单电子近似， 电子平均场,1. Hartree方程(1928),连乘积形式：,按变分原理， 的选取E达到极小,正交归一条件,单电子方程,Hartree方程中的势：,第二项是全部电子在r处形成的势，与 相抵消 第三项是须扣除的自作用，与j有关，但如取r为计算原点：,所以对凝胶模型，Hartree方程：,相互作用没有相互作用 电子正电荷背景自由电子气,3. Hartree-Fock方程
2、(1930) Hartree方程不满足Pauli不相容原理 电子：费米子 单电子波函数f：N电子 体系的总波函数：,不涉及自旋轨道耦合时：,N电子体系能量期待值：,1.第二项j,j可以相等，自相互作用 2.自相互作用严格相消（通过第二，三项） 3.第三项为交换项，同自旋电子,通过变分:,么正变换:,单电子方程：,与Hartree方程的差别：第三项对全体电子，第四项新增，交换作用项。求和只涉及与j态自旋平行的j态，是电子服从Fermi统计的反映。,4. Koopmann定理（1934),单电子轨道能量等于N电子体系从第j个轨道上取走一个电子并保持N1个电子状态不不变的总能变化值。,推广：系统中一
3、个电子由状态j转移到态i而引起系统能量的变化,5. 交换空穴(Fermi hole) 将H-F方程改写为：,其中：,定性讨论：假设,Fermi hole:与某电子自旋相同的其余邻近电子在围绕该电子形成总量为1的密度亏欠域,energy as a function of the one electron density, nuclear-electron attraction, electron-electron repulsion Thomas-Fermi approximation for the kinetic energy Slater approximation for the exc
4、hange energy,6. 密度泛函理论(Density functional theory),(1) Thomas-Fermi-Dirac Model,(2) The Hohenberg-Kohn Theorem,properties are uniquely determined by the ground-state electron,In 1964, Hohenberg and Kohn proved that,molecular energy, wave function and all other molecular electronic,probability density
5、,namely,Phys. Rev. 136, 13864 (1964),.”,Density functional theory (DFT) attempts to,and other ground-state molecular properties,from the ground-state electron density,“For molecules with a nondegenerate ground state, the ground-state,calculate,Proof:,The electronic Hamiltonian is,it is produced by c
6、harges external to the system of electrons.,In DFT,is called the external potential acting on electron i, since,Once the external potential,the electronic wave functions and allowed energies of the molecule are,and the number of electrons n are specified,determined as the solutions of the electronic
7、 Schrdinger equation.,Now we need to prove that the ground-state electron probability density,the number of electrons.,the external potential (except for an arbitrary additive constant),a) Since,determines the number of electrons.,b) To see that,determines the external potential, we suppose,that thi
8、s is false and that there are two external potentials,and,(differing,by more than a constant) that each give rise to the same ground-state electron,density,.,determines,the exact ground-state wave function and energy of,the exact ground-state wave function and energy of,Let,Since,and,differ by more
9、than a constant,and,must be different functions.,Proof:,Assume,thus,thus,which contradicts the given,information.,function, the exact ground-state wave function,state energy,for a given Hamiltonian,If the ground state is nondegenerate, then there is only one normalized,that gives the exact ground,Ac
10、cording to the variation theorem, suppose that,If,then,is any normalized,well-behaved trial variation function.,Now use,as a trial function with the Hamiltonian,then,Substituting,gives,Let,be a function of the spatial coordinates,of electron i,then,Using the above result, we get,Similarly, if we go
11、through the same reasoning with a and b interchanged,we get,By hypothesis, the two different wave functions give the same electron,. Putting,and adding the above two inequalities,density:,yield,potentials could produce the same ground-state electron density must be false.,energy) and also determines
12、 the number of electrons.,This result is false, so our initial assumption that two different external,potential (to within an additive constant that simply affects the zero level of,Hence, the ground-state electron probability density,determines the external,probability density,and other properties”
13、,emphasizes the dependence of,the external potential,differs for different molecules.,“For systems with a nondegenerate ground state, the ground-state electron,determines the ground-state wave function and energy, which,However, the functionals,are unknown.,is also written as,The functional,independ
14、ent of the external,on,is,potential.,(3) The Hohenberg-kohn variational theorem,“For every trial density function,that satisfies,and,for all, the following inequality holds:,is the true groundstate energy.”,Proof:,Let,satisfy that,and,Hohenberg-Kohn theorem,determines the external potential,and this
15、 in turn determines the wave function,density,. By the,that corresponds to the,.,where,with Hamiltonian,. According to the variation theorem,Let us use the wave function,as a trial variation function for the molecule,Since the left hand side of this inequality can be rewritten as,One gets,states. Su
16、bsequently, Levy proved the theorems for degenerate ground states.,Hohenberg and Kohn proved their theorems only for nondegenerate ground,(4) The Kohn-Sham method,If we know the ground-state electron density,molecular properties from,function., the Hohenberg-Kohn,theorem tells us that it is possible in principle to calculate all the ground-state, without having to find the molecular wave,1965, Kohn and Sham devised a practical method for finding,and,for fin
