吀栀椀猀 猀攀挀琀椀漀渀 椀猀 搀攀猀椀最渀攀搀 琀漀 猀攀爀瘀攀 愀猀 愀 猀琀愀爀琀椀渀最 瀀漀椀渀琀 昀漀爀 愀挀焀甀椀爀椀渀最 愀 眀漀爀欀椀渀最 欀渀漀眀氀攀搀最攀 漀昀 琀栀攀 挀甀爀爀攀渀琀 最攀渀攀爀愀琀椀漀渀 漀昀 焀甀愀渀琀甀洀 挀栀攀洀椀挀愀氀 洀攀琀栀漀搀猀⸀ 吀栀攀 攀洀瀀栀愀猀椀猀 椀渀 琀栀椀猀 眀漀爀欀 椀猀 漀渀 琀栀攀 爀攀氀愀琀椀漀渀猀栀椀瀀猀 戀攀琀眀攀攀渀 椀洀瀀漀爀琀愀渀琀 焀甀愀渀琀甀洀 挀栀攀洀椀挀愀氀 挀漀渀挀攀瀀琀猀 愀渀搀 琀栀攀椀爀 洀愀琀栀攀洀愀琀椀挀愀氀 昀漀甀渀搀愀琀椀漀渀猀⸀ 吀栀椀猀 瀀爀攀猀攀渀琀愀琀椀漀渀 椀猀 氀漀漀猀攀氀礀 戀愀猀攀搀 漀渀 䰀漀眀攀⸀
匀琀愀渀搀椀渀最 圀愀瘀攀猀 椀渀 愀 䌀氀愀洀瀀攀搀 匀琀爀椀渀最㰀⼀瀀㸀㰀倀㸀 The properties of waves in a string clamped at both ends (clamped string) are analogous to some of the important basic quantum mechanical properties of atoms and molecules. Waves can be generated by plucking (adding energy to) such a string:㰀⼀瀀㸀㰀倀㸀 By observation, waves in a clamped string must adopt discrete wavelengths because the ends of the string cannot be displaced.
Such waves can be mathematically described by solving the wave equation for . The one-dimensional form of the wave equation is sufficient to describe the clamped string model:㰀⼀瀀㸀㰀倀㸀 The solutions to the wave equation, , are called the wavefunctions. is proportional to the energy density at any point along the wave (i.e. the amount of energy from the pluck that is stored in each part of the string).㰀⼀瀀㸀㰀倀㸀 All solutions to the wave equation must be symmetric. However, there are two possible kinds of wave symmetries. The first type of symmetry is such that the wave and its reflection about the wave coordinate axis are superimposable (symmetric):㰀⼀瀀㸀㰀倀㸀 The second type of symmetry is such that the wave and its reflection are not superimposable (antisymmetric):
An asymmetric waveform is shown below for comparison purposes: 圀愀瘀攀ⴀ䰀椀欀攀 倀愀爀琀椀挀氀攀猀 ⠀琀栀攀 搀攀 䈀爀漀最氀椀攀 圀愀瘀攀氀攀渀最琀栀⤀ The idea that particles have wave-like properties was developed by de Broglie. Planck had shown in 1900 that electromagnetic radiation was emitted and absorbed in discrete quanta having energy proportional to the frequency of the radiation . Einstein showed in 1905 that the energy of a particle is , where m is the particle's relativistic mass. Combining these two expressions yields the relationship between a photon's momentum and frequency (or wavelength):
That is, electromagnetic radiation has particle-like characteristics (momentum) in addition to its wave-like characteristics (wavelength, diffraction). In 1924 de Broglie suggested that matter also had this dual nature and proposed that a wavelength can be associated with the momentum of any particle, not just photons: As a particle's momentum becomes large (i.e. due to its mass), becomes undetectably small. It is useful to express in terms of energy:㰀⼀瀀㸀㰀倀㸀 Schrodinger's Equation for a One-Electron Atom吀栀攀 挀漀渀渀攀挀琀椀漀渀 戀攀琀眀攀攀渀 挀氀愀猀猀椀挀愀氀 眀愀瘀攀猀 愀渀搀 搀攀 䈀爀漀最氀椀攀✀猀 瀀愀爀琀椀挀氀攀 眀愀瘀攀猀 眀愀猀 洀愀搀攀 戀礀 匀挀栀爀漀搀椀渀最攀爀⸀ 吀栀攀 挀氀愀猀猀椀挀愀氀 琀栀爀攀攀ⴀ搀椀洀攀渀猀椀漀渀愀氀 眀愀瘀攀 攀焀甀愀琀椀漀渀 椀猀㨀 㰀⼀瀀㸀㰀倀㸀 眀栀攀爀攀 椀猀 琀栀攀 䰀愀瀀氀愀挀椀愀渀 漀瀀攀爀愀琀漀爀 ⠀⤀ 愀渀搀 椀猀 琀栀攀 眀愀瘀攀昀甀渀挀琀椀漀渀 搀攀猀挀爀椀戀椀渀最 琀栀攀 搀椀猀瀀氀愀挀攀洀攀渀琀 愀琀 愀渀礀 瀀漀椀渀琀 愀氀漀渀最 琀栀攀 眀愀瘀攀⸀ 匀挀栀爀漀搀椀渀最攀爀 猀甀戀猀琀椀琀甀琀攀搀 琀栀攀 搀攀 䈀爀漀最氀椀攀 眀愀瘀攀氀攀渀最琀栀 昀漀爀 琀漀 愀搀愀瀀琀 琀栀攀 挀氀愀猀猀椀挀愀氀 眀愀瘀攀 攀焀甀愀琀椀漀渀 琀漀 瀀愀爀琀椀挀氀攀 眀愀瘀攀猀⸀ 吀栀攀 匀挀栀爀漀搀椀渀最攀爀 攀焀甀愀琀椀漀渀 椀猀 最椀瘀攀渀 戀礀㨀
吀栀椀猀 攀焀甀愀琀椀漀渀 挀愀渀 戀攀 爀攀愀爀爀愀渀最攀搀 椀渀 愀 猀攀爀椀攀猀 漀昀 愀氀最攀戀爀愀椀挀 猀琀攀瀀猀 琀漀 愀 洀漀爀攀 挀漀渀瘀攀渀椀攀渀琀 昀漀爀洀㨀 㰀⼀瀀㸀㰀倀㸀 䠀 椀猀 挀愀氀氀攀搀 琀栀攀 䠀愀洀椀氀琀漀渀椀愀渀 漀瀀攀爀愀琀漀爀⸀ 匀甀戀猀琀椀琀甀琀椀渀最 䠀 椀渀琀漀 匀挀栀爀漀搀椀渀最攀爀✀猀 攀焀甀愀琀椀漀渀 氀攀愀搀猀 琀漀㨀 椀猀 挀愀氀氀攀搀 愀渀 攀椀最攀渀昀甀渀挀琀椀漀渀 愀渀搀 䔀 愀渀 攀椀最攀渀瘀愀氀甀攀⸀ 吀栀攀 䠀愀洀椀氀琀漀渀椀愀渀 伀瀀攀爀愀琀漀爀 The electrostatic potential energy, V, of one charged particle in the field of another is given by:㰀⼀瀀㸀㰀倀㸀 where q is the charge on each particle and r is the distance between them. If q1 and q2 are both + or both -, they repel (V > 0). If q1 and q2 are opposite, they attract (V < 0). The potential energy between an electron (q = -e) and a nucleus (q = +Ze) can be written as: Substituting for V in the wave equation gives:
V decreases as the electron and nucleus get closer. So, the electron moves faster when it's close to the nucleus, and slower when it's farther away (because energy is interconverted from T into V to maintain constant E). The second derivative of is the rate of change of slope of at any given point, which describes the curvature of "wiggliness" of the function. So, the wigglier is at a given point in space, the greater is the kinetic energy of the electron at that point.㰀⼀瀀㸀㰀倀㸀 Properties of and E䔀愀挀栀 愀渀搀 椀琀猀 挀漀爀爀攀猀瀀漀渀搀椀渀最 攀渀攀爀最礀 爀攀氀愀琀攀 琀漀 愀 猀椀渀最氀攀 攀氀攀挀琀爀漀渀 戀漀甀渀搀 琀漀 愀 渀甀挀氀攀甀猀 漀昀 挀栀愀爀最攀 ⬀娀攀⸀ 吀栀攀 愀爀攀 挀愀氀氀攀搀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 漀爀戀椀琀愀氀猀 ⠀漀昀琀攀渀 挀愀氀氀攀搀 栀礀搀爀漀最攀渀ⴀ氀椀欀攀 愀琀漀洀椀挀 漀爀戀椀琀愀氀猀⤀⸀ 吀栀攀礀 愀爀攀 搀攀猀椀最渀愀琀攀搀 愀猀 猀Ⰰ ㈀猀Ⰰ ㈀瀀Ⰰ 攀琀挀⸀
吀栀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 愀琀漀洀椀挀 漀爀戀椀琀愀氀猀 愀爀攀 漀戀琀愀椀渀攀搀 戀礀 愀渀愀氀礀琀椀挀愀氀氀礀 猀漀氀瘀椀渀最 匀挀栀爀漀搀椀渀最攀爀✀猀 攀焀甀愀琀椀漀渀⸀ 最椀瘀攀猀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 搀攀渀猀椀琀礀 漀昀 昀椀渀搀椀渀最 愀渀 攀氀攀挀琀爀漀渀 愀琀 愀 最椀瘀攀渀 瀀漀椀渀琀 椀渀 猀瀀愀挀攀⸀ 䔀愀挀栀 挀愀渀 戀攀 挀漀渀琀漀甀爀攀搀 ⠀椀⸀攀⸀ 最爀愀瀀栀攀搀 愀琀 愀 挀漀渀猀琀愀渀琀 瘀愀氀甀攀⤀⸀ 一漀琀攀 琀栀愀琀 攀愀挀栀 猀漀氀甀琀椀漀渀 栀愀猀 愀 爀愀搀椀愀氀 琀攀爀洀 挀漀渀琀愀椀渀椀渀最 爀Ⰰ 愀渀搀 猀漀洀攀 栀愀瘀攀 愀渀最甀氀愀爀 琀攀爀洀猀 挀漀渀琀愀椀渀椀渀最 愀渀搀 ⸀ 吀栀攀 猀 漀爀戀椀琀愀氀猀 愀爀攀 猀瀀栀攀爀椀挀愀氀Ⰰ 愀渀搀 琀栀攀爀攀昀漀爀攀 栀愀瘀攀 渀漀 愀渀最甀氀愀爀 搀攀瀀攀渀搀攀渀挀攀⸀ 䄀氀氀 漀琀栀攀爀 漀爀戀椀琀愀氀猀 栀愀瘀攀 猀瀀愀琀椀愀氀 搀椀爀攀挀琀椀漀渀愀氀椀琀礀Ⰰ 愀渀搀 琀栀攀 挀漀爀爀攀猀瀀漀渀搀椀渀最 眀愀瘀攀昀甀渀挀琀椀漀渀猀 栀愀瘀攀 愀渀最甀氀愀爀 琀攀爀洀猀⸀
䌀氀愀猀猀椀挀愀氀氀礀Ⰰ 愀渀 攀氀攀挀琀爀漀渀 愀渀搀 渀甀挀氀攀甀猀 愀爀攀 戀漀甀渀搀 戀礀 愀渀 愀琀琀爀愀挀琀椀瘀攀 瀀漀琀攀渀琀椀愀氀 琀栀愀琀 瘀愀爀椀攀猀 眀椀琀栀 琀栀攀椀爀 搀椀猀琀愀渀挀攀Ⰰ 爀⸀ 吀 椀猀 渀攀瘀攀爀 猀甀昀昀椀挀椀攀渀琀氀礀 氀愀爀最攀 昀漀爀 愀渀 攀氀攀挀琀爀漀渀 琀漀 攀猀挀愀瀀攀 琀栀椀猀 瀀漀琀攀渀琀椀愀氀⸀ 吀栀甀猀Ⰰ 嘀 搀攀昀椀渀攀猀 愀 氀漀眀攀爀 戀漀甀渀搀 琀漀 䔀 ⠀椀⸀攀⸀ 䔀 㸀 嘀 昀漀爀 愀氀氀 爀⤀⸀
唀渀搀攀爀 焀甀愀渀琀甀洀 洀攀挀栀愀渀椀挀猀Ⰰ 栀漀眀攀瘀攀爀Ⰰ 攀氀攀挀琀爀漀渀猀 挀愀渀 攀砀椀猀琀 愀琀 氀愀爀最攀爀 搀椀猀琀愀渀挀攀猀 琀栀愀渀 愀氀氀漀眀攀搀 戀礀 嘀 ⠀椀⸀攀⸀ 䔀 㰀 嘀⤀⸀ 琀愀瀀攀爀猀 漀昀昀 攀砀瀀漀渀攀渀琀椀愀氀氀礀 漀甀琀猀椀搀攀 漀昀 琀栀椀猀 爀愀渀最攀 ⠀椀⸀攀⸀ 挀漀渀琀愀椀渀猀 愀渀 椀洀愀最椀渀愀爀礀 挀漀洀瀀漀渀攀渀琀⤀⸀ 吀栀椀猀 椀猀 挀愀氀氀攀搀 ∀琀甀渀渀攀氀椀渀最∀⸀ 愀渀搀 愀氀氀 挀栀攀洀椀挀愀氀 瀀爀漀瀀攀爀琀椀攀猀 搀攀爀椀瘀攀搀 琀栀攀爀攀昀爀漀洀 愀爀攀 爀攀愀氀 瘀愀氀甀攀搀Ⰰ 愀渀搀 猀漀 椀猀 最椀瘀攀渀 戀礀 Ⰰ 眀栀攀爀攀 椀猀 琀栀攀 挀漀洀瀀氀攀砀 挀漀渀樀甀最愀琀攀 漀昀 ⸀
吀栀攀 猀甀洀 漀昀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 搀攀渀猀椀琀礀 漀瘀攀爀 琀栀攀 攀渀琀椀爀攀 瘀漀氀甀洀攀 漀昀 猀瀀愀挀攀 洀甀猀琀 爀攀昀氀攀挀琀 琀栀攀 攀砀椀猀琀攀渀挀攀 漀昀 琀栀攀 攀氀攀挀琀爀漀渀 猀漀洀攀眀栀攀爀攀 椀渀 猀瀀愀挀攀 ⠀渀漀爀洀愀氀椀稀愀琀椀漀渀 挀漀渀搀椀琀椀漀渀⤀㨀
眀栀攀爀攀 搀瘀 椀猀 愀渀 椀渀昀椀渀椀琀攀猀椀洀愀氀 挀栀甀渀欀 漀昀 瘀漀氀甀洀攀 愀渀搀 琀栀攀 挀✀猀 愀爀攀 渀漀爀洀愀氀椀稀愀琀椀漀渀 挀漀渀猀琀愀渀琀猀 琀栀愀琀 愀搀樀甀猀琀 琀栀攀 琀漀 猀愀琀椀猀昀礀 琀栀攀 渀漀爀洀愀氀椀稀愀琀椀漀渀 攀焀甀愀琀椀漀渀⸀ 挀愀渀 戀攀 猀礀洀洀攀琀爀椀挀 漀爀 愀渀琀椀猀礀洀洀攀琀爀椀挀Ⰰ 戀甀琀 渀攀瘀攀爀 甀渀猀礀洀洀攀琀爀椀挀⸀ 吀栀攀 猀甀瀀攀爀瀀漀猀椀琀椀漀渀 漀昀 愀氀氀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 漀昀 愀渀 愀琀漀洀 椀猀 猀瀀栀攀爀椀挀愀氀氀礀 猀礀洀洀攀琀爀椀挀 ⠀椀⸀攀⸀ 琀栀攀爀攀 椀猀 渀漀 昀愀瘀漀爀攀搀 搀椀爀攀挀琀椀漀渀 椀渀 愀渀 愀琀漀洀⤀⸀ 䔀愀挀栀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀 挀愀渀 攀砀椀猀琀 椀渀 琀眀漀 昀漀爀洀猀 ⠀ 愀渀搀 ⤀ 挀愀氀氀攀搀 ∀猀瀀椀渀∀ 猀琀愀琀攀猀⸀ 䤀渀搀椀瘀椀搀甀愀氀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 漀渀 琀栀攀 猀愀洀攀 愀琀漀洀 搀漀 渀漀琀 漀瘀攀爀氀愀瀀㨀 㰀⼀瀀㸀㰀倀㸀 吀栀攀 猀漀氀甀琀椀漀渀猀 昀漀爀 䔀 愀爀攀 搀椀猀挀爀攀琀攀 ⠀焀甀愀渀琀椀稀攀搀⤀Ⰰ 栀愀瘀椀渀最 渀 㴀 Ⰰ㈀Ⰰ㌀⸀⸀⸀ 瀀漀猀猀椀戀氀攀 瘀愀氀甀攀猀Ⰰ 爀攀洀椀渀椀猀挀攀渀琀 漀昀 琀栀攀 渀 㴀 Ⰰ㈀Ⰰ㌀⸀⸀⸀ 眀愀瘀攀氀攀渀最琀栀猀 ⠀最椀瘀攀渀 戀礀 ㈀䰀⼀渀⤀ 漀昀 愀 挀氀愀洀瀀攀搀 猀琀爀椀渀最⸀ 匀挀栀爀漀搀椀渀最攀爀✀猀 䔀焀甀愀琀椀漀渀 昀漀爀 愀 䴀甀氀琀椀ⴀ䔀氀攀挀琀爀漀渀 䄀琀漀洀 The Hamiltonian for an atom with k electrons is given by:
The Schrodinger equation for a multi-electron atom can be solved numerically, although Velectron-electron cannot be included as an explicit term in the Hamiltonian. Rather, its effect on can be accounted for by a mathematically simpler approach: that each electron interacts with an average of the nucleus + all other electrons (self-consistent field approximation). Neglect of Electron-Electron Repulsion and Separability of the Hamiltonian䌀漀渀猀椀搀攀爀 愀 琀眀漀ⴀ攀氀攀挀琀爀漀渀 愀琀漀洀 ⠀攀⸀最⸀ 䠀攀⤀ 椀渀 眀栀椀挀栀 嘀攀氀攀挀琀爀漀渀ⴀ攀氀攀挀琀爀漀渀 椀猀 渀攀最氀攀挀琀攀搀㨀 眀栀攀爀攀 栀⠀椀⤀ 愀爀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 䠀愀洀椀氀琀漀渀椀愀渀猀⸀ 匀挀栀爀漀搀椀渀最攀爀✀猀 攀焀甀愀琀椀漀渀 挀愀渀 琀栀攀渀 戀攀 愀瀀瀀爀漀砀椀洀愀琀攀搀 愀猀㨀 㰀⼀瀀㸀㰀倀㸀 眀栀攀爀攀 愀渀搀 愀爀攀 琀栀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 攀渀攀爀最椀攀猀 猀甀挀栀 琀栀愀琀 ⠀琀栀攀 琀漀琀愀氀 攀渀攀爀最礀⤀ 愀渀搀 椀猀 挀漀洀瀀漀猀攀搀 漀昀 愀 挀漀洀戀椀渀愀琀椀漀渀 漀昀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀猀Ⰰ ⸀ 匀椀渀挀攀 攀氀攀挀琀爀漀渀ⴀ攀氀攀挀琀爀漀渀 椀渀琀攀爀愀挀琀椀漀渀猀 愀爀攀 渀攀最氀攀挀琀攀搀 椀渀 琀栀椀猀 愀瀀瀀爀漀愀挀栀Ⰰ 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 昀椀渀搀椀渀最 攀椀琀栀攀爀 攀氀攀挀琀爀漀渀 愀琀 愀 最椀瘀攀渀 瀀漀猀椀琀椀漀渀 ⠀椀⸀攀⸀ ⤀ 搀漀攀猀 渀漀琀 搀攀瀀攀渀搀 漀渀 琀栀攀 瀀漀猀椀琀椀漀渀 漀昀 琀栀攀 漀琀栀攀爀 攀氀攀挀琀爀漀渀⸀ 吀栀椀猀 挀愀渀 戀攀 洀愀琀栀攀洀愀琀椀挀愀氀氀礀 爀攀瀀爀攀猀攀渀琀攀搀 甀猀椀渀最 琀栀攀 瀀爀漀搀甀挀琀猀 漀昀 琀栀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀猀㨀 㰀⼀瀀㸀㰀倀㸀 吀栀攀 愀爀最甀洀攀渀琀猀 漀昀 琀栀攀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 搀攀渀漀琀攀 琀栀攀 瀀漀猀椀琀椀漀渀猀 漀昀 攀愀挀栀 攀氀攀挀琀爀漀渀⸀ 吀栀攀 洀甀氀琀椀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀 洀甀猀琀 琀愀欀攀 椀渀琀漀 挀漀渀猀椀搀攀爀愀琀椀漀渀 琀栀攀 昀愀挀琀 琀栀愀琀 攀氀攀挀琀爀漀渀猀 愀爀攀 椀渀搀椀猀琀椀渀最甀椀猀栀愀戀氀攀Ⰰ 愀渀搀 琀栀攀爀攀昀漀爀攀 椀渀琀攀爀挀栀愀渀最椀渀最 攀氀攀挀琀爀漀渀 瀀漀猀椀琀椀漀渀 愀猀猀椀最渀洀攀渀琀猀 椀渀 愀 眀愀瘀攀昀甀渀挀琀椀漀渀 挀愀渀渀漀琀 氀攀愀搀 琀漀 愀 搀椀昀昀攀爀攀渀琀 眀愀瘀攀昀甀渀挀琀椀漀渀⸀ 吀栀椀猀 椀猀 渀漀琀 琀栀攀 挀愀猀攀 昀漀爀 琀栀攀 瀀爀漀搀甀挀琀猀 琀栀攀洀猀攀氀瘀攀猀㨀 㰀⼀瀀㸀㰀倀㸀 戀甀琀 椀猀 琀爀甀攀 漀昀 琀栀攀 猀甀洀 漀昀 瀀攀爀洀甀琀攀搀 瀀爀漀搀甀挀琀猀㨀 愀渀搀 琀栀攀椀爀 搀椀昀昀攀爀攀渀挀攀㨀 眀栀攀爀攀 愀渀搀 愀爀攀 琀栀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 ⠀攀⸀最⸀ 猀Ⰰ ㈀猀Ⰰ 攀琀挀⸀⤀⸀ 䤀渀琀攀爀挀栀愀渀最椀渀最 攀氀攀挀琀爀漀渀猀 椀渀 琀栀攀 琀眀漀 攀焀甀愀琀椀漀渀猀 氀攀愀搀猀 琀漀㨀 愀渀搀 一漀琀攀 琀栀愀琀 琀栀攀 猀椀最渀 挀栀愀渀最攀猀 昀漀爀 眀栀攀渀 攀氀攀挀琀爀漀渀 瀀漀猀椀琀椀漀渀 愀猀猀椀最渀洀攀渀琀猀 愀爀攀 椀渀琀攀爀挀栀愀渀最攀搀Ⰰ 戀甀琀 琀栀攀 攀焀甀愀琀椀漀渀猀 愀爀攀 漀琀栀攀爀眀椀猀攀 甀渀愀昀昀攀挀琀攀搀 戀礀 琀栀椀猀 漀瀀攀爀愀琀椀漀渀⸀ 愀渀搀 愀爀攀 挀愀氀氀攀搀 ∀猀瀀愀挀攀∀ 昀甀渀挀琀椀漀渀猀 戀攀挀愀甀猀攀 琀栀攀礀 搀攀瀀攀渀搀 漀渀 琀栀攀 猀瀀愀琀椀愀氀 瀀漀猀椀琀椀漀渀猀 漀昀 琀栀攀 攀氀攀挀琀爀漀渀猀⸀ 匀瀀椀渀 Electrons can exist in two possible states called "spin" states designated as and . Spin states are mathematically represented by "spin" functions, and , analogous to and . Spin functions must be symmetric or antisymmetric with respect to the interchange of electron state assignments. For a two electron system, the possible spin functions are: A complete wavefunction, , is composed of a space function ( or ) and a spin function ( or ): In practice, only the antisymmetric form, , is physically meaningful. results from symmetrically opposite constituent space and spin functions: and There are four possible antisymmetric atomic spin orbitals for a 1s(1) 2s(2) configuration: 㰀⼀瀀㸀㰀倀㸀 where the are approximate wavefunctions for the entire atom, made by "mixing" one-electron spin orbitals. Antisymmetric spin orbitals are often constructed using a mathematical function called a Slater determinant. Electron Correlation Effects嘀攀氀攀挀琀爀漀渀ⴀ攀氀攀挀琀爀漀渀 眀愀猀 渀攀最氀攀挀琀攀搀 椀渀 琀栀攀 䠀愀洀椀氀琀漀渀椀愀渀 椀渀 琀栀攀 瀀爀攀瘀椀漀甀猀 猀攀挀琀椀漀渀猀⸀ 吀栀攀 攀昀昀攀挀琀猀 漀昀 攀氀攀挀琀爀漀渀ⴀ攀氀攀挀琀爀漀渀 椀渀琀攀爀愀挀琀椀漀渀猀 愀爀攀Ⰰ 椀渀 最攀渀攀爀愀氀Ⰰ 挀愀氀氀攀搀 ∀攀氀攀挀琀爀漀渀 挀漀爀爀攀氀愀琀椀漀渀∀⸀ 愀渀搀 䔀 挀愀渀渀漀琀 戀攀 甀猀攀搀 琀漀 挀漀爀爀攀挀琀氀礀 瀀爀攀搀椀挀琀 愀琀漀洀椀挀 瀀爀漀瀀攀爀琀椀攀猀 眀椀琀栀漀甀琀 猀漀洀攀栀漀眀 愀挀挀漀甀渀琀椀渀最 昀漀爀 攀氀攀挀琀爀漀渀 挀漀爀爀攀氀愀琀椀漀渀⸀ 䴀甀氀琀椀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 ⠀⤀ 愀爀攀 漀瘀攀爀氀礀 椀渀昀氀甀攀渀挀攀搀 戀礀 渀甀挀氀攀愀爀ⴀ攀氀攀挀琀爀漀渀 愀琀琀爀愀挀琀椀漀渀Ⰰ 愀渀搀 琀栀椀猀 琀攀渀搀猀 琀漀 猀瀀愀琀椀愀氀氀礀 挀漀渀琀爀愀挀琀 琀栀攀 攀氀攀挀琀爀漀渀 搀攀渀猀椀琀礀 搀椀猀琀爀椀戀甀琀椀漀渀 琀漀眀愀爀搀 琀栀攀 渀甀挀氀攀甀猀⸀ 吀栀椀猀 椀猀 挀漀渀琀爀愀爀礀 琀漀 琀栀攀 甀渀愀挀挀漀甀渀琀攀搀 攀昀昀攀挀琀 漀昀 攀氀攀挀琀爀漀渀ⴀ攀氀攀挀琀爀漀渀 爀攀瀀甀氀猀椀漀渀Ⰰ 眀栀椀挀栀 琀攀渀搀猀 琀漀 洀愀欀攀 琀栀攀 漀爀戀椀琀愀氀猀 氀愀爀最攀爀 愀渀搀 洀漀爀攀 搀椀昀昀甀猀攀⸀ 吀栀攀 漀瘀攀爀攀猀琀椀洀愀琀攀搀 搀攀最爀攀攀 漀昀 挀漀渀琀爀愀挀琀椀漀渀 挀愀渀 戀攀 挀漀爀爀攀挀琀攀搀 戀礀 椀洀瀀氀椀挀椀琀氀礀 爀攀搀甀挀椀渀最 琀栀攀 攀昀昀攀挀琀猀 漀昀 渀甀挀氀攀愀爀ⴀ攀氀攀挀琀爀漀渀 愀琀琀爀愀挀琀椀漀渀⸀ 吀栀攀 爀愀琀椀漀渀愀氀攀 椀猀 琀栀愀琀 攀愀挀栀 攀氀攀挀琀爀漀渀 椀猀 猀挀爀攀攀渀攀搀 昀爀漀洀 琀栀攀 渀甀挀氀攀甀猀 戀礀 琀栀攀 漀琀栀攀爀 攀氀攀挀琀爀漀渀猀⸀ 吀栀攀 猀挀爀攀攀渀椀渀最 攀昀昀攀挀琀 椀猀 愀猀猀甀洀攀搀 琀漀 戀攀 愀瘀攀爀愀最攀搀 漀瘀攀爀 愀氀氀 漀琀栀攀爀 攀氀攀挀琀爀漀渀猀⸀ 吀栀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 ⠀⤀ 挀愀渀 戀攀 椀洀瀀爀漀瘀攀搀 椀渀 琀栀椀猀 爀攀最愀爀搀 戀礀 洀漀搀椀昀礀椀渀最 琀栀攀 渀甀挀氀攀愀爀 挀栀愀爀最攀 挀漀渀猀琀愀渀琀Ⰰ 娀㨀 㰀⼀瀀㸀㰀倀㸀 挀愀渀 戀攀 眀爀椀琀琀攀渀 愀猀㨀 眀栀攀爀攀 椀猀 愀渀 愀搀樀甀猀琀愀戀氀攀 瀀愀爀愀洀攀琀攀爀⸀ 䄀猀 戀攀挀漀洀攀猀 猀洀愀氀氀攀爀Ⰰ 琀栀攀 漀爀戀椀琀愀氀 琀攀渀搀猀 琀漀 攀砀瀀愀渀搀⸀ 挀愀渀 戀攀 搀攀琀攀爀洀椀渀攀搀 甀猀椀渀最 琀栀攀 猀攀氀昀ⴀ挀漀渀猀椀猀琀攀渀琀 昀椀攀氀搀 愀瀀瀀爀漀愀挀栀 ⠀匀䌀䘀⤀⸀ 䤀渀 瀀爀愀挀琀椀挀攀Ⰰ 洀漀爀攀 甀猀攀昀甀氀 昀漀爀洀猀 漀昀 愀爀攀 愀瘀愀椀氀愀戀氀攀 ⠀攀⸀最⸀ 匀氀愀琀攀爀 愀渀搀 䜀愀甀猀猀椀愀渀ⴀ琀礀瀀攀 漀爀戀椀琀愀氀猀⤀⸀ 吀栀攀 匀挀栀爀漀搀椀渀最攀爀 䔀焀甀愀琀椀漀渀 昀漀爀 愀 䴀漀氀攀挀甀氀攀 The Hamiltonian for a molecule with N atoms and k electrons is given by:
Note that Tnucleus is omitted. This (Born-Oppenheimer) approximation considers the nuclei to be stationary relative to the electron motions. Since the positions of the nuclei are fixed during the calculations, Vnucleus-nucleus is typically treated as a constant. Just as for multi-electron atoms, approximate molecular wavefunctions (molecular spin orbitals) can be created using a set of one-electron wavefunctions. The major differences between the various molecular orbital calculation methods pertains to their consideration (or lack of consideration) of electron correlation. Basis Sets吀栀攀 猀攀琀 漀昀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 甀猀攀搀 琀漀 戀甀椀氀搀 洀漀氀攀挀甀氀愀爀 漀爀戀椀琀愀氀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 椀猀 挀愀氀氀攀搀 琀栀攀 戀愀猀椀猀 猀攀琀⸀ 吀栀攀 栀礀搀爀漀最攀渀ⴀ氀椀欀攀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 洀漀搀椀昀椀攀搀 昀漀爀 攀氀攀挀琀爀漀渀 挀漀爀爀攀氀愀琀椀漀渀 愀爀攀 最攀渀攀爀愀氀氀礀 渀漀琀 甀猀攀搀 瀀攀爀 猀攀 戀攀挀愀甀猀攀 琀栀攀礀 氀攀愀搀 琀漀 洀愀琀栀攀洀愀琀椀挀愀氀 挀漀洀瀀氀椀挀愀琀椀漀渀猀 愀渀搀 琀椀洀攀ⴀ挀漀渀猀甀洀椀渀最 挀愀氀挀甀氀愀琀椀漀渀猀⸀ 䤀渀猀琀攀愀搀Ⰰ 眀愀瘀攀昀甀渀挀琀椀漀渀猀 愀爀攀 甀猀攀搀 椀渀 眀栀椀挀栀 琀栀攀 爀愀搀椀愀氀 琀攀爀洀猀Ⰰ Ⰰ 愀爀攀 猀椀洀瀀氀椀昀椀攀搀⸀ 吀栀攀 洀漀猀琀 挀漀洀洀漀渀 猀甀挀栀 眀愀瘀攀昀甀渀挀琀椀漀渀猀 愀爀攀 匀氀愀琀攀爀ⴀ琀礀瀀攀 漀爀戀椀琀愀氀猀 ⠀匀吀伀⤀㨀 㰀⼀瀀㸀㰀倀㸀 眀栀攀爀攀 猀 椀猀 愀 猀挀爀攀攀渀椀渀最 挀漀渀猀琀愀渀琀Ⰰ 愀渀搀 娀 ⴀ 猀 挀愀渀 戀攀 琀愀欀攀渀 愀猀 愀渀搀 䜀愀甀猀猀椀愀渀ⴀ琀礀瀀攀 漀爀戀椀琀愀氀猀 ⠀䜀吀伀⤀㨀 眀栀攀爀攀 椀猀 愀 挀甀爀瘀攀ⴀ昀椀琀琀椀渀最 挀漀渀猀琀愀渀琀 甀猀攀搀 琀漀 愀瀀瀀爀漀砀椀洀愀琀攀 愀渀 匀吀伀⸀ 吀栀攀 匀吀伀 猀挀爀攀攀渀椀渀最 挀漀渀猀琀愀渀琀猀 愀爀攀 挀愀氀挀甀氀愀琀攀搀 昀漀爀 猀洀愀氀氀 洀漀搀攀氀 洀漀氀攀挀甀氀攀猀 甀猀椀渀最 爀椀最漀爀漀甀猀 猀攀氀昀ⴀ挀漀渀猀椀猀琀攀渀琀 昀椀攀氀搀 洀攀琀栀漀搀猀Ⰰ 愀渀搀 琀栀攀渀 最攀渀攀爀愀氀椀稀攀搀 昀漀爀 甀猀攀 眀椀琀栀 愀挀琀甀愀氀 洀漀氀攀挀甀氀攀猀 漀昀 椀渀琀攀爀攀猀琀⸀ 吀栀椀猀 搀愀琀愀 椀猀 猀甀瀀瀀氀椀攀搀 眀椀琀栀 琀栀攀 瘀愀爀椀漀甀猀 猀漀昀琀眀愀爀攀 椀洀瀀氀攀洀攀渀琀愀琀椀漀渀猀 琀栀愀琀 甀猀攀 匀吀伀猀⸀ 吀栀攀 洀愀琀栀攀洀愀琀椀挀愀氀 爀攀焀甀椀爀攀洀攀渀琀猀 昀漀爀 猀漀氀瘀椀渀最 琀栀攀 椀渀琀攀最爀愀氀猀 漀昀 琀栀攀 眀愀瘀攀 攀焀甀愀琀椀漀渀 甀猀椀渀最 匀吀伀猀 愀爀攀 琀椀洀攀 挀漀渀猀甀洀椀渀最⸀ 吀栀攀 愀挀挀甀爀愀挀礀 漀昀 匀吀伀猀 挀愀渀 戀攀 椀洀瀀爀漀瘀攀搀 戀礀 挀漀洀戀椀渀椀渀最 琀眀漀 漀爀 洀漀爀攀 匀吀伀猀 ⠀椀⸀攀⸀ 眀椀琀栀 琀眀漀 搀椀昀昀攀爀攀渀琀 瘀愀氀甀攀猀 漀昀 ⤀ 椀渀琀漀 愀 猀椀渀最氀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 眀愀瘀攀昀甀渀挀琀椀漀渀 ⠀搀漀甀戀氀攀ⴀ 戀愀猀椀猀 猀攀琀⤀㨀 ऀ㰀⼀瀀㸀㰀倀㸀 䰀椀欀攀眀椀猀攀Ⰰ 匀吀伀猀 挀愀渀 戀攀 搀攀瘀椀猀攀搀 琀栀愀琀 爀攀昀氀攀挀琀 琀栀攀 猀栀愀瀀攀 瀀爀漀瀀攀爀琀椀攀猀 漀昀 瀀漀氀愀爀椀稀攀搀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 漀爀戀椀琀愀氀猀 ⠀攀⸀最⸀ 琀栀攀 挀漀洀戀椀渀愀琀椀漀渀 漀昀 愀 猀 愀渀搀 愀 ㈀瀀稀 漀爀戀椀琀愀氀⤀⸀ 䜀愀甀猀猀椀愀渀 漀爀戀椀琀愀氀猀 愀爀攀 洀愀琀栀攀洀愀琀椀挀愀氀氀礀 猀椀洀瀀氀攀爀 琀栀愀渀 匀吀伀猀Ⰰ 戀甀琀 氀攀猀猀 愀挀挀甀爀愀琀攀⸀ 䄀 猀 䜀吀伀 愀渀搀 琀栀攀 挀漀爀爀攀猀瀀漀渀搀椀渀最 漀渀攀ⴀ攀氀攀挀琀爀漀渀 栀礀搀爀漀最攀渀ⴀ氀椀欀攀 漀爀戀椀琀愀氀 愀爀攀 挀漀洀瀀愀爀攀搀 椀渀 琀栀攀 昀漀氀氀漀眀椀渀最 瀀氀漀琀㨀
ऀ 䄀氀氀 漀昀 琀栀攀 漀渀攀ⴀ攀氀攀挀琀爀漀渀 漀爀戀椀琀愀氀猀 挀愀渀 戀攀 戀甀椀氀琀 戀礀 挀漀洀戀椀渀椀渀最 猀攀琀猀 漀昀 最愀甀猀猀椀愀渀 昀甀渀挀琀椀漀渀猀 ⠀最愀甀猀猀椀愀渀 瀀爀椀洀椀琀椀瘀攀猀⤀ 琀栀愀琀 愀瀀瀀爀漀砀椀洀愀琀攀 攀愀挀栀 匀吀伀⸀ 吀栀攀 爀攀猀甀氀琀 椀猀 挀愀氀氀攀搀 愀 挀漀渀琀爀愀挀琀攀搀 最愀甀猀猀椀愀渀 昀甀渀挀琀椀漀渀⸀ 䄀 洀椀渀椀洀愀氀 戀愀猀椀猀 猀攀琀 椀猀 漀渀攀 椀渀 眀栀椀挀栀 漀渀氀礀 漀挀挀甀瀀椀攀搀 漀爀戀椀琀愀氀猀 漀昀 攀愀挀栀 椀猀漀氀愀琀攀搀 愀琀漀洀 愀爀攀 甀猀攀搀 琀漀 挀漀洀瀀漀猀攀 琀栀攀 洀漀氀攀挀甀氀愀爀 漀爀戀椀琀愀氀猀⸀ 唀渀漀挀挀甀瀀椀攀搀 洀漀氀攀挀甀氀愀爀 漀爀戀椀琀愀氀猀 愀爀攀 挀愀氀氀攀搀 瘀椀爀琀甀愀氀 漀爀戀椀琀愀氀猀⸀ 䄀搀搀椀琀椀漀渀愀氀 椀渀昀漀爀洀愀琀椀漀渀 漀渀 戀愀猀椀猀 猀攀琀猀 椀猀 愀瘀愀椀氀愀戀氀攀 ⠀䨀愀渀 䰀愀戀愀渀漀眀猀欀椀Ⰰ 伀栀椀漀 匀甀瀀攀爀挀漀洀瀀甀琀攀爀 䌀攀渀琀攀爀⤀⸀ 䄀 氀椀戀爀愀爀礀 漀昀 戀愀猀椀猀 猀攀琀猀 椀猀 洀愀椀渀琀愀椀渀攀搀 戀礀 琀栀攀 䔀渀瘀椀爀漀渀洀攀渀琀愀氀 䴀漀氀攀挀甀氀愀爀 匀挀椀攀渀挀攀猀 䰀愀戀漀爀愀琀漀爀礀 ⠀倀愀挀椀昀椀挀 一漀爀琀栀眀攀猀琀 䰀愀戀漀爀愀琀漀爀礀Ⰰ 䈀愀琀琀攀氀氀攀 䴀攀洀漀爀椀愀氀 䤀渀猀琀椀琀甀琀攀⤀⸀ 㰀⼀瀀㸀㰀倀㸀 My early life was spent in Burnham-on-Sea, Somerset, a small seaside resort town (population around 5000) on the west coast of England. 䤀 眀愀猀 戀漀爀渀 漀渀 伀挀琀漀戀攀爀 ㌀Ⰰ 㤀㈀㔀 愀渀搀 氀椀瘀攀搀 琀栀攀爀攀 眀椀琀栀 洀礀 瀀愀爀攀渀琀猀 甀渀琀椀氀 猀栀漀爀琀氀礀 愀昀琀攀爀 琀栀攀 攀渀搀 漀昀 琀栀攀 匀攀挀漀渀搀 圀漀爀氀搀 圀愀爀 椀渀 㤀㐀㘀⸀ 一漀 洀攀洀戀攀爀 漀昀 洀礀 昀愀洀椀氀礀 眀愀猀 椀渀瘀漀氀瘀攀搀 椀渀 愀渀礀 猀挀椀攀渀琀椀昀椀挀 漀爀 琀攀挀栀渀椀挀愀氀 愀挀琀椀瘀椀琀礀⸀ 䤀渀搀攀攀搀Ⰰ 䤀 眀愀猀 琀栀攀 昀椀爀猀琀 琀漀 愀琀琀攀渀搀 愀 甀渀椀瘀攀爀猀椀琀礀⸀
䴀礀 昀愀琀栀攀爀Ⰰ 䬀攀椀琀栀 倀漀瀀氀攀Ⰰ 漀眀渀攀搀 琀栀攀 瀀爀椀渀挀椀瀀愀氀 洀攀渀✀猀 挀氀漀琀栀椀渀最 猀琀漀爀攀 椀渀 䈀甀爀渀栀愀洀⸀ 䤀渀 愀搀搀椀琀椀漀渀 琀漀 猀攀氀氀椀渀最 挀氀漀琀栀攀猀 椀渀 琀栀攀 猀栀漀瀀Ⰰ 栀攀 甀猀攀搀 琀漀 搀爀椀瘀攀 愀爀漀甀渀搀 琀栀攀 猀甀爀爀漀甀渀搀椀渀最 挀漀甀渀琀爀礀猀椀搀攀 眀椀琀栀 愀 挀愀爀 昀甀氀氀 漀昀 挀氀漀琀栀攀猀 昀漀爀 瀀攀漀瀀氀攀 椀渀 爀攀洀漀琀攀 昀愀爀洀猀 愀渀搀 瘀椀氀氀愀最攀猀⸀ 䠀攀 眀愀猀 爀攀猀漀甀爀挀攀昀甀氀 愀渀搀 洀愀搀攀 愀 昀愀椀爀 椀渀挀漀洀攀Ⰰ 挀漀渀猀椀搀攀爀椀渀最 琀栀攀 攀挀漀渀漀洀椀挀 搀椀昀昀椀挀甀氀琀椀攀猀 搀甀爀椀渀最 琀栀攀 搀攀瀀爀攀猀猀椀漀渀 漀昀 琀栀攀 㤀㌀ 猀⸀ 䴀礀 最爀攀愀琀ⴀ最爀愀渀搀昀愀琀栀攀爀 栀愀搀 挀漀洀攀 琀漀 䈀甀爀渀栀愀洀 愀爀漀甀渀搀 㠀㔀 愀渀搀 猀攀琀 甀瀀 愀 渀甀洀戀攀爀 漀昀 氀漀挀愀氀 戀甀猀椀渀攀猀猀攀猀⸀ 䠀攀 栀愀搀 愀 氀愀爀最攀 昀愀洀椀氀礀 愀渀搀 琀栀攀猀攀 眀攀爀攀 猀瀀氀椀琀 甀瀀 愀洀漀渀最 栀椀猀 挀栀椀氀搀爀攀渀⸀ 䄀猀 愀 爀攀猀甀氀琀Ⰰ 䤀 栀愀搀 爀攀氀愀琀椀瘀攀猀 椀渀 洀愀渀礀 漀昀 琀栀攀 漀琀栀攀爀 戀甀猀椀渀攀猀猀攀猀 椀渀 琀栀攀 琀漀眀渀⸀ 䴀礀 最爀愀渀搀昀愀琀栀攀爀 椀渀栀攀爀椀琀攀搀 琀栀攀 挀氀漀琀栀椀渀最 猀栀漀瀀 愀渀搀 琀栀椀猀 瀀愀猀猀攀搀 琀漀 洀礀 昀愀琀栀攀爀 眀栀攀渀 栀攀 爀攀琀甀爀渀攀搀 昀爀漀洀 琀栀攀 愀爀洀礀 愀琀 攀渀搀 漀昀 琀栀攀 䘀椀爀猀琀 圀漀爀氀搀 圀愀爀⸀
䴀礀 洀漀琀栀攀爀Ⰰ 䴀愀爀礀 䨀漀渀攀猀Ⰰ 挀愀洀攀 昀爀漀洀 愀 昀愀爀洀椀渀最 戀愀挀欀最爀漀甀渀搀⸀ 䠀攀爀 昀愀琀栀攀爀 栀愀搀 洀漀瘀攀搀 昀爀漀洀 匀栀爀漀瀀猀栀椀爀攀 愀猀 愀 礀漀甀渀最 洀愀渀 愀渀搀 栀愀搀 昀愀爀洀攀搀 渀攀愀爀 䈀愀琀栀 昀漀爀 洀漀猀琀 漀昀 栀椀猀 氀椀昀攀⸀ 䤀 猀甀猀瀀攀挀琀 琀栀愀琀 栀攀 眀漀甀氀搀 栀愀瘀攀 瀀爀攀昀攀爀爀攀搀 琀漀 戀攀 愀 琀攀愀挀栀攀爀Ⰰ 昀漀爀 栀攀 栀愀搀 愀 氀愀爀最攀 挀漀氀氀攀挀琀椀漀渀 漀昀 戀漀漀欀猀 愀渀搀 攀渀挀礀挀氀漀瀀攀搀椀愀猀⸀ 䠀攀 眀愀渀琀攀搀 洀礀 洀漀琀栀攀爀 琀漀 戀攀 愀 猀挀栀漀漀氀琀攀愀挀栀攀爀Ⰰ 戀甀琀 琀栀椀猀 搀椀搀 渀漀琀 栀愀瀀瀀攀渀⸀ 䤀渀猀琀攀愀搀Ⰰ 猀栀攀 戀攀挀愀洀攀 愀 琀甀琀漀爀 琀漀 挀栀椀氀搀爀攀渀 椀渀 愀 爀椀挀栀 昀愀洀椀氀礀 愀渀搀Ⰰ 氀愀琀攀爀Ⰰ 愀 氀椀戀爀愀爀椀愀渀 椀渀 琀栀攀 愀爀洀礀 搀甀爀椀渀最 琀栀攀 昀椀爀猀琀 眀愀爀⸀ 䴀漀猀琀 漀昀 栀攀爀 爀攀氀愀琀椀瘀攀猀 眀攀爀攀 昀愀爀洀攀爀猀 椀渀 瘀愀爀椀漀甀猀 瀀愀爀琀猀 漀昀 匀漀洀攀爀猀攀琀 愀渀搀 圀椀氀琀猀栀椀爀攀 猀漀Ⰰ 愀猀 猀洀愀氀氀 挀栀椀氀搀爀攀渀Ⰰ 洀礀 礀漀甀渀最攀爀 戀爀漀琀栀攀爀 愀渀搀 䤀 猀瀀攀渀琀 洀甀挀栀 琀椀洀攀 猀琀愀礀椀渀最 漀渀 昀愀爀洀猀⸀
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䄀琀 琀栀攀 愀最攀 漀昀 琀眀攀氀瘀攀Ⰰ 䤀 搀攀瘀攀氀漀瀀攀搀 愀渀 椀渀琀攀渀猀攀 椀渀琀攀爀攀猀琀 椀渀 洀愀琀栀攀洀愀琀椀挀猀⸀ 伀渀 攀砀瀀漀猀甀爀攀 琀漀 愀氀最攀戀爀愀Ⰰ 䤀 眀愀猀 昀愀猀挀椀渀愀琀攀搀 戀礀 猀椀洀甀氀琀愀渀攀漀甀猀 攀焀甀愀琀椀漀渀猀 愀渀搀 爀愀瀀椀搀氀礀 爀攀愀搀 愀栀攀愀搀 漀昀 琀栀攀 挀氀愀猀猀 琀漀 琀栀攀 攀渀搀 漀昀 琀栀攀 戀漀漀欀⸀ 䤀 昀漀甀渀搀 愀 搀椀猀挀愀爀搀攀搀 琀攀砀琀戀漀漀欀 漀渀 挀愀氀挀甀氀甀猀 椀渀 愀 眀愀猀琀攀戀愀猀欀攀琀 愀渀搀 爀攀愀搀 椀琀 昀爀漀洀 挀漀瘀攀爀 琀漀 挀漀瘀攀爀⸀ 圀椀琀栀椀渀 愀 礀攀愀爀Ⰰ 䤀 眀愀猀 昀愀洀椀氀椀愀爀 眀椀琀栀 洀漀猀琀 漀昀 琀栀攀 渀漀爀洀愀氀 猀挀栀漀漀氀 洀愀琀栀攀洀愀琀椀挀愀氀 挀甀爀爀椀挀甀氀甀洀⸀ 䤀 攀瘀攀渀 猀琀愀爀琀攀搀 猀漀洀攀 爀攀猀攀愀爀挀栀 瀀爀漀樀攀挀琀猀Ⰰ 昀漀爀洀甀氀愀琀椀渀最 琀栀攀 琀栀攀漀爀礀 漀昀 瀀攀爀洀甀琀愀琀椀漀渀猀 椀渀 爀攀猀瀀漀渀猀攀 琀漀 愀 挀栀愀氀氀攀渀最攀 愀戀漀甀琀 琀栀攀 渀甀洀戀攀爀 漀昀 瀀漀猀猀椀戀氀攀 戀愀琀琀椀渀最 漀爀搀攀爀猀 漀昀 琀栀攀 攀氀攀瘀攀渀 瀀氀愀礀攀爀猀 椀渀 愀 挀爀椀挀欀攀琀 琀攀愀洀⸀ 䘀漀爀 愀 瘀攀爀礀 猀栀漀爀琀 琀椀洀攀Ⰰ 䤀 琀栀漀甀最栀琀 琀栀椀猀 琀漀 戀攀 漀爀椀最椀渀愀氀 眀漀爀欀 戀甀琀 眀愀猀 洀漀爀琀椀昀椀攀搀 琀漀 昀椀渀搀 渀℀ 搀攀猀挀爀椀戀攀搀 椀渀 愀 琀攀砀琀戀漀漀欀⸀ 䤀 琀栀攀渀 愀琀琀攀洀瀀琀攀搀 琀漀 攀砀琀攀渀搀 渀℀ 琀漀 昀爀愀挀琀椀漀渀愀氀 渀甀洀戀攀爀猀 戀礀 瘀愀爀椀漀甀猀 椀渀琀攀爀瀀漀氀愀琀椀漀渀 猀挀栀攀洀攀猀⸀ 䐀攀猀瀀椀琀攀 愀 氀漀琀 漀昀 攀昀昀漀爀琀Ⰰ 琀栀椀猀 瀀爀漀樀攀挀琀 眀愀猀 甀氀琀椀洀愀琀攀氀礀 甀渀猀甀挀挀攀猀猀昀甀氀㬀 䤀 眀愀猀 愀渀最爀礀 眀椀琀栀 洀礀猀攀氀昀 眀栀攀渀 䤀 氀攀愀爀渀攀搀 漀昀 䔀甀氀攀爀✀猀 猀漀氀甀琀椀漀渀 猀漀洀攀 礀攀愀爀猀 氀愀琀攀爀⸀ 䠀漀眀攀瘀攀爀Ⰰ 琀栀攀猀攀 攀愀爀氀礀 攀砀瀀攀爀椀攀渀挀攀猀 眀攀爀攀 瘀愀氀甀愀戀氀攀 椀渀 昀漀爀洀甀氀愀琀椀渀最 愀渀 愀琀琀椀琀甀搀攀 漀昀 瀀攀爀猀椀猀琀攀渀挀攀 椀渀 爀攀猀攀愀爀挀栀⸀
䄀氀氀 琀栀椀猀 洀愀琀栀攀洀愀琀椀挀愀氀 愀挀琀椀瘀椀琀礀 眀愀猀 欀攀瀀琀 猀攀挀爀攀琀⸀ 䴀礀 瀀愀爀攀渀琀猀 搀椀搀 渀漀琀 挀漀洀瀀爀攀栀攀渀搀 眀栀愀琀 䤀 眀愀猀 搀漀椀渀最 愀渀搀Ⰰ 椀渀 挀氀愀猀猀Ⰰ 䤀 漀昀琀攀渀 椀渀琀爀漀搀甀挀攀搀 搀攀氀椀戀攀爀愀琀攀 攀爀爀漀爀猀 椀渀 洀礀 攀砀攀爀挀椀猀攀猀 琀漀 愀瘀漀椀搀 最椀瘀椀渀最 愀渀 椀洀瀀爀攀猀猀椀漀渀 漀昀 戀攀椀渀最 琀漀漀 挀氀攀瘀攀爀⸀ 䴀礀 最爀愀搀攀猀 漀甀琀猀椀搀攀 漀昀 洀愀琀栀攀洀愀琀椀挀猀 愀渀搀 猀挀椀攀渀挀攀 眀攀爀攀 甀渀搀椀猀琀椀渀最甀椀猀栀攀搀 猀漀 䤀 甀猀甀愀氀氀礀 攀渀搀攀搀 甀瀀 猀攀瘀攀爀愀氀 瀀氀愀挀攀猀 搀漀眀渀 椀渀 琀栀攀 洀漀渀琀栀氀礀 挀氀愀猀猀 漀爀搀攀爀⸀ 吀栀椀猀 愀氀氀 挀栀愀渀最攀搀 猀甀搀搀攀渀氀礀 琀栀爀攀攀 礀攀愀爀猀 氀愀琀攀爀 眀栀攀渀 琀栀攀 渀攀眀 猀攀渀椀漀爀 洀愀琀栀攀洀愀琀椀挀猀 琀攀愀挀栀攀爀Ⰰ 刀⸀䌀⸀ 䰀礀渀攀猀猀Ⰰ 搀攀挀椀搀攀搀 琀漀 挀栀愀氀氀攀渀最攀 琀栀攀 挀氀愀猀猀 眀椀琀栀 愀渀 甀渀甀猀甀愀氀氀礀 搀椀昀昀椀挀甀氀琀 琀攀猀琀⸀ 䤀 猀甀挀挀甀洀戀攀搀 琀漀 琀攀洀瀀琀愀琀椀漀渀 愀渀搀 琀甀爀渀攀搀 椀渀 愀 瀀攀爀昀攀挀琀 瀀愀瀀攀爀Ⰰ 眀椀琀栀 洀甀氀琀椀瀀氀攀 猀漀氀甀琀椀漀渀猀 琀漀 洀愀渀礀 漀昀 琀栀攀 瀀爀漀戀氀攀洀猀⸀ 匀栀漀爀琀氀礀 愀昀琀攀爀眀愀爀搀猀Ⰰ 洀礀 瀀愀爀攀渀琀猀 愀渀搀 䤀 眀攀爀攀 猀甀洀洀漀渀攀搀 琀漀 愀 猀瀀攀挀椀愀氀 挀漀渀昀攀爀攀渀挀攀 眀椀琀栀 琀栀攀 栀攀愀搀洀愀猀琀攀爀 愀琀 眀栀椀挀栀 椀琀 眀愀猀 搀攀挀椀搀攀搀 琀栀愀琀 䤀 猀栀漀甀氀搀 戀攀 瀀爀攀瀀愀爀攀搀 昀漀爀 愀 猀挀栀漀氀愀爀猀栀椀瀀 椀渀 洀愀琀栀攀洀愀琀椀挀猀 愀琀 䌀愀洀戀爀椀搀最攀 唀渀椀瘀攀爀猀椀琀礀⸀ 䐀甀爀椀渀最 琀栀攀 爀攀洀愀椀渀椀渀最 琀眀漀 礀攀愀爀猀 愀琀 䈀䜀匀Ⰰ 䤀 爀攀挀攀椀瘀攀搀 椀渀琀攀渀猀攀 瀀攀爀猀漀渀愀氀 挀漀愀挀栀椀渀最 昀爀漀洀 䰀礀渀攀猀猀 愀渀搀 琀栀攀 猀攀渀椀漀爀 瀀栀礀猀椀挀猀 洀愀猀琀攀爀Ⰰ 吀⸀䄀⸀ 䴀漀爀爀椀猀⸀ 䈀漀琀栀 眀攀爀攀 漀甀琀猀琀愀渀搀椀渀最 琀攀愀挀栀攀爀猀⸀ 吀栀攀 猀挀栀漀漀氀Ⰰ 氀椀欀攀 洀愀渀礀 漀琀栀攀爀猀 椀渀 䈀爀椀琀愀椀渀Ⰰ 愀琀琀愀挀栀攀搀 最爀攀愀琀 椀洀瀀漀爀琀愀渀挀攀 琀漀 琀栀攀 瀀氀愀挀攀洀攀渀琀 漀昀 猀琀甀搀攀渀琀猀 愀琀 伀砀昀漀爀搀 漀爀 䌀愀洀戀爀椀搀最攀⸀ 䴀漀猀琀 猀甀挀栀 愀眀愀爀搀猀 眀攀爀攀 椀渀 琀栀攀 挀氀愀猀猀椀挀猀 愀渀搀 䤀 琀栀椀渀欀 琀栀愀琀 琀栀攀 洀愀琀栀攀洀愀琀椀挀猀 愀渀搀 猀挀椀攀渀挀攀 猀琀愀昀昀 眀攀爀攀 瘀攀爀礀 愀渀砀椀漀甀猀 琀漀 挀漀洀瀀攀琀攀⸀ 䤀爀漀渀椀挀愀氀氀礀Ⰰ 搀甀爀椀渀最 琀栀攀 氀愀猀琀 琀眀漀 礀攀愀爀猀 愀琀 䈀䜀匀Ⰰ 䤀 愀戀愀渀搀漀渀攀搀 挀栀攀洀椀猀琀爀礀 琀漀 挀漀渀挀攀渀琀爀愀琀攀 漀渀 洀愀琀栀攀洀愀琀椀挀猀 愀渀搀 瀀栀礀猀椀挀猀⸀ 䤀渀 㤀㐀㈀Ⰰ 䤀 琀爀愀瘀攀氀氀攀搀 琀漀 䌀愀洀戀爀椀搀最攀 琀漀 琀愀欀攀 琀栀攀 猀挀栀漀氀愀爀猀栀椀瀀 攀砀愀洀椀渀愀琀椀漀渀 愀琀 吀爀椀渀椀琀礀 䌀漀氀氀攀最攀Ⰰ 爀攀挀攀椀瘀攀搀 愀渀 愀眀愀爀搀 愀渀搀 攀渀琀攀爀攀搀 琀栀攀 甀渀椀瘀攀爀猀椀琀礀 椀渀 伀挀琀漀戀攀爀 㤀㐀㌀⸀
䤀渀 琀栀攀 洀椀搀搀氀攀 漀昀 琀栀攀 眀愀爀Ⰰ 洀漀猀琀 礀漀甀渀最 洀攀渀 漀昀 洀礀 愀最攀 眀攀爀攀 椀渀搀甀挀琀攀搀 椀渀琀漀 琀栀攀 愀爀洀攀搀 昀漀爀挀攀猀 愀琀 琀栀攀 愀最攀 漀昀 猀攀瘀攀渀琀攀攀渀⸀ 䠀漀眀攀瘀攀爀Ⰰ 愀 猀洀愀氀氀 最爀漀甀瀀 漀昀 猀琀甀搀攀渀琀猀 椀渀 洀愀琀栀攀洀愀琀椀挀猀Ⰰ 猀挀椀攀渀挀攀 愀渀搀 洀攀搀椀挀椀渀攀 眀愀猀 瀀攀爀洀椀琀琀攀搀 琀漀 愀琀琀攀渀搀 甀渀椀瘀攀爀猀椀琀礀 戀攀昀漀爀攀 琀愀欀椀渀最 瀀愀爀琀 椀渀 眀愀爀琀椀洀攀 爀攀猀攀愀爀挀栀 瀀爀漀樀攀挀琀猀 猀甀挀栀 愀猀 爀愀搀愀爀Ⰰ 渀甀挀氀攀愀爀 攀砀瀀氀漀猀椀瘀攀猀Ⰰ 挀漀搀攀ⴀ戀爀攀愀欀椀渀最 愀渀搀 琀栀攀 氀椀欀攀⸀ 吀栀椀猀 眀愀猀 愀 栀椀最栀氀礀 猀甀挀挀攀猀猀昀甀氀 瀀爀漀樀攀挀琀 愀渀搀 洀愀渀礀 漀昀 洀礀 瀀爀攀搀攀挀攀猀猀漀爀猀 椀渀 攀愀爀氀椀攀爀 礀攀愀爀猀 洀愀搀攀 椀洀瀀漀爀琀愀渀琀 挀漀渀琀爀椀戀甀琀椀漀渀猀 琀漀 琀栀攀 眀愀爀 攀昀昀漀爀琀⸀ 吀栀攀 瀀氀愀渀 眀愀猀 琀漀 挀漀洀瀀氀攀琀攀 愀氀氀 搀攀最爀攀攀 挀漀甀爀猀攀猀 椀渀 漀渀氀礀 琀眀漀 礀攀愀爀猀Ⰰ 昀漀氀氀漀眀攀搀 戀礀 猀攀挀漀渀搀洀攀渀琀 琀漀 愀 最漀瘀攀爀渀洀攀渀琀 爀攀猀攀愀爀挀栀 攀猀琀愀戀氀椀猀栀洀攀渀琀⸀ 䤀渀 洀礀 挀愀猀攀Ⰰ 䤀 挀漀洀瀀氀攀琀攀搀 倀愀爀琀 䤀䤀 漀昀 琀栀攀 洀愀琀栀攀洀愀琀椀挀愀氀 琀爀椀瀀漀猀 椀渀 䴀愀礀 㤀㐀㔀Ⰰ 樀甀猀琀 愀猀 琀栀攀 䔀甀爀漀瀀攀愀渀 眀愀爀 眀愀猀 攀渀搀椀渀最⸀ 䤀渀 昀愀挀琀Ⰰ 椀琀 眀愀猀 栀愀爀搀 琀漀 挀漀渀挀攀渀琀爀愀琀攀 漀渀 琀栀攀 攀砀愀洀椀渀愀琀椀漀渀猀 戀攀挀愀甀猀攀 漀昀 琀栀攀 渀漀椀猀礀 挀攀氀攀戀爀愀琀椀漀渀猀 最漀椀渀最 漀渀 椀渀 琀栀攀 猀琀爀攀攀琀猀 漀甀琀猀椀搀攀⸀ 吀栀攀 最漀瘀攀爀渀洀攀渀琀 渀漀 氀漀渀最攀爀 栀愀搀 渀攀攀搀 昀漀爀 洀礀 猀攀爀瘀椀挀攀猀 愀渀搀 琀栀攀 甀渀椀瘀攀爀猀椀琀礀 眀愀猀 甀渀搀攀爀 最爀攀愀琀 瀀爀攀猀猀甀爀攀 琀漀 洀愀欀攀 爀漀漀洀 昀漀爀 琀栀攀 搀攀氀甀最攀 漀昀 攀砀猀攀爀瘀椀挀攀洀攀渀 愀猀 琀栀攀礀 眀攀爀攀 搀攀洀漀戀椀氀椀稀攀搀 昀爀漀洀 琀栀攀 愀爀洀攀搀 昀漀爀挀攀猀⸀ 匀漀Ⰰ 䤀 栀愀搀 琀漀 氀攀愀瘀攀 䌀愀洀戀爀椀搀最攀 愀渀搀 琀愀欀攀 甀瀀 椀渀搀甀猀琀爀椀愀氀 攀洀瀀氀漀礀洀攀渀琀 昀漀爀 愀 瀀攀爀椀漀搀⸀ 吀栀椀猀 眀愀猀 眀椀琀栀 琀栀攀 䈀爀椀猀琀漀氀 䄀攀爀漀瀀氀愀渀攀 䌀漀洀瀀愀渀礀Ⰰ 挀氀漀猀攀 琀漀 眀栀攀爀攀 䤀 栀愀搀 愀琀琀攀渀搀攀搀 猀挀栀漀漀氀⸀ 吀栀攀爀攀 眀愀猀 氀椀琀琀氀攀 琀漀 搀漀 琀栀攀爀攀 愀渀搀 䤀 栀愀搀 愀 瀀攀爀椀漀搀 漀昀 攀渀昀漀爀挀攀搀 椀搀氀攀渀攀猀猀 愀猀 挀栀愀渀最椀渀最 攀洀瀀氀漀礀洀攀渀琀 眀愀猀 椀氀氀攀最愀氀 愀琀 琀栀攀 琀椀洀攀 ⠀瀀愀爀琀 漀昀 琀栀攀 漀戀猀攀猀猀椀漀渀 昀漀爀 愀 瀀氀愀渀渀攀搀 攀挀漀渀漀洀礀 椀渀 瀀漀猀琀眀愀爀 䈀爀椀琀愀椀渀⤀⸀
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