1、量子化學量子化學 第八章第八章樊建芬樊建芬Chapter 8 Introduction of Density Function Theory第八章第八章 密度泛函實際簡介密度泛函實際簡介量子化學量子化學 第八章第八章 從從20世紀世紀60年代密度泛函實際年代密度泛函實際DFT提出以提出以來，并在局域密度近似下導出著名的來，并在局域密度近似下導出著名的Kohn-Sham方程以來，方程以來，DFT已逐漸成為量子化學計算領域的強已逐漸成為量子化學計算領域的強有力的工具。有力的工具。 在量子化學計算領域，據在量子化學計算領域，據INSPEC數據記錄顯數據記錄顯示示:直至直至80年代末，分子軌道年代末，

2、分子軌道HF方法不斷占主導位方法不斷占主導位置；置；9090年代中期，年代中期，DFTDFT和和HFHF方法的論文并駕齊驅；之方法的論文并駕齊驅；之 后，后，DFTDFT的任務按指數級數添加。的任務按指數級數添加。量子化學量子化學 第八章第八章 最近幾年，密度泛函方法得到了廣泛的運用，最近幾年，密度泛函方法得到了廣泛的運用，通常可以得到比通常可以得到比HFHF方法更好的計算結果，但只需中方法更好的計算結果，但只需中等程度的價錢，對于中型或大型分子體系，其計算等程度的價錢，對于中型或大型分子體系，其計算本錢遠低于本錢遠低于MP2MP2法。法。 2019 2019年，年，KohnKohn因因DFT

3、DFT的開創性任務與的開創性任務與PoplePople共同共同獲得諾貝爾化學獎。從中可見獲得諾貝爾化學獎。從中可見DFTDFT在化學領域中的在化學領域中的位置。位置。量子化學量子化學 第八章第八章電子動電子動能算符能算符核電子核電子吸引能吸引能電子電子電子電子排斥能排斥能電子相關電子相關交換能交換能+ =E 注：電子相關能：主要包括自旋注：電子相關能：主要包括自旋軌道巧協作用能、軌道巧協作用能、 自旋自旋自旋巧協作用能、軌道自旋巧協作用能、軌道軌道巧協作用能。軌道巧協作用能。 電子交換能，電子具有全同粒子特性，滿足保電子交換能，電子具有全同粒子特性，滿足保里原理，波函數必需具有反對稱性，它與非

4、對稱里原理，波函數必需具有反對稱性，它與非對稱波函數的能量差為交換能。波函數的能量差為交換能。分子體系的分子體系的SchrSchrdingerdinger方程方程23量子化學量子化學 第八章第八章兩種形狀的波函數不同，能量也有所差別。兩種形狀的波函數不同，能量也有所差別。 例：例：Li+的某一激發態的某一激發態1s12s1, 假設電子排布形狀為假設電子排布形狀為1s 2s1 (1) (1) 2 (2) (2)ss為非對稱波函數為非對稱波函數1 (1) (1)2 (1) (1)11 (2) (2)2 (2) (2)2ssss為反對稱波函數為反對稱波函數量子化學量子化學 第八章第八章 但是它在描畫

5、化學鍵時有些缺乏，而且沒有相關但是它在描畫化學鍵時有些缺乏，而且沒有相關能校正時不宜處置熱化學問題。能校正時不宜處置熱化學問題。 HF實際可以在分子尺寸提供較準確的交換能處置，實際可以在分子尺寸提供較準確的交換能處置，同時也是大體系的實踐計算工具。同時也是大體系的實踐計算工具。 HF超自洽場的校正，如超自洽場的校正，如MP多級微擾、組態相互多級微擾、組態相互作用作用CI也只能處置中小分子，對大體系是不適用的。也只能處置中小分子，對大體系是不適用的。量子化學量子化學 第八章第八章量子化學量子化學 第八章第八章量子化學量子化學 第八章第八章 Density functional theory (D

6、FT) attempts to calculate and other ground-state molecular properties from the ground-state electron density 0E.0Proof:The electronic Hamiltonian is000HE external potential11目錄量子化學量子化學 第八章第八章 In DFT, is called the external potential acting on electron i, since it is produced by charges external to t

7、he system of electrons. Once the external potential and the number of electrons n are specified, the electronic wave functions and allowed energies of the molecule are determined as the solutions of the electronic Schrdinger equation. 量子化學量子化學 第八章第八章(1) the external potential (except for an arbitrar

8、y additive constant) ; It can be proved that the ground-state electron probability density determines: nrdr)(0(2) the number of electrons.9量子化學量子化學 第八章第八章 “For systems with a nondegenerate ground state, the ground-state electron probability density determines the ground-state wave function and energ

9、y, and other properties。 emphasizes the dependence of on the external potential , which differs for different molecules.0E00EE 量子化學量子化學 第八章第八章Here, However, the functionalsare unknown. 量子化學量子化學 第八章第八章is also written as0E0000( ) ( )EErr drFHere,000eeFTV The functional is independent of the external p

10、otential. 0F目錄 量子化學量子化學 第八章第八章8.2 The Hohenberg-kohn variational theorem the following inequality holds: where is the true groundstate energy.0trEE0E “For every trial density function that satisfies and for all , ( )trr( )0trr drr目錄量子化學量子化學 第八章第八章 Let satisfy that and . By the Hohenberg-Kohn theorem

11、, determines the external potential and this in turn determines the wave function that corresponds to the density . Proof: 量子化學量子化學 第八章第八章 Let us use the wave function as a trial variation function for the molecule with Hamiltonian . According to the variation theorem 量子化學量子化學 第八章第八章Since the left h

12、and side of this inequality can be rewritten as Hohenberg and Kohn proved their theorems only for nondegenerate ground states. Subsequently, Levy proved the theorems for degenerate ground states. ( )trtrtrtreeTVr drE One gets 0trEE目錄 量子化學量子化學 第八章第八章8.3 The Kohn-Sham method If we know the ground-stat

13、e electron density , the Hohenberg-Kohn theorem tells us that it is possible in principle to calculate all the ground-state molecular properties from , without having to find the molecular wave function.4量子化學量子化學 第八章第八章 Their method is capable, in principle, of yielding exact results, but because th

14、e equations of the Kohn-Sham (KS) method contain an unknown functional that must be approximated, the KS formation of DFT yield approximate results. 1965, Kohn and Sham devised a practical method for finding and for finding from . Phys. Rev., 140, A 1133 (1965). 量子化學量子化學 第八章第八章. Kohn and Sham consid

15、ered a fictitious reference system s of n noninteracting electrons that each experience the same external potential that makes the ground-state electron probability density of the reference system equal to the exact of the molecule we are interested in: 量子化學量子化學 第八章第八章 Since the electrons do not int

16、eract with one another in the reference system, the Hamiltonian of the reference system is where is the one-electron Kohn-Sham Hamiltonian. 量子化學量子化學 第八章第八章 The KS operator is the same as the HF operator except that the exchange operators in the HF operator are replaced by , which handles the effects

17、 of both exchange and electron correlation. 量子化學量子化學 第八章第八章 Various approximate functionals are used in molecular DFT calculations. The functional is written as the sum of an exchange-energy functional and a correlation-energy functional :4 量子化學量子化學 第八章第八章 Among various approximations, gradient-corr

18、ected exchange and correlation energy functionals are the most accurate. Commonly used and . 量子化學量子化學 第八章第八章Lee-Yang-Parr (LYP) functionalP86 (the Perdew 1986 correlation functional) PW91 (Perdew and Wangs 1991 functional)PW86 (Perdew and Wangs 1986 functional)B88 (Beckes 1988 functional)目錄量子化學量子化學 第八章第八章8.4 Comments on the DFT methods(1) The KS equations are solved in a self-consistent fashion, like the HF equations.(2) The computation time required for a DFT calculation formally scales the third power of the number of basis functions.(3