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Summary Chapter 2 1. Solution for H-like atom/ion (one-electron system) radial & angular functions of Atomic orbitals, electron cloud, quantum numbers.

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Presentation on theme: "Summary Chapter 2 1. Solution for H-like atom/ion (one-electron system) radial & angular functions of Atomic orbitals, electron cloud, quantum numbers."— Presentation transcript:

1 Summary Chapter 2 1. Solution for H-like atom/ion (one-electron system) radial & angular functions of Atomic orbitals, electron cloud, quantum numbers (n, l, m, and ms) electronic orbital/spin angular momentum & space quantization. Atomic-orbital wavefunctions are eigenfunctions of such operators as H, L2, & Lz!

2 Key points: For H-like atom/ions, the wavefunctions to describe their atomic orbitals (AOs) that are derived from their their Schrodinger equations can be characterized using three quantum numbers, i.e., n, l, m, and symbolized as nlm. They are eigenfunctions of those hermitian operators Ĥ, L2 (M2) and Lz (Mz), by the following eigenequations, ^ Magnitude of orbital angular momentum Note: Angular momentum L is a vector!

3 Chapter 2 Summary Solution for many-electron atom
a) Separation of variables Independent particle approximation (mean field model)  i) one-particle eigenequations ii) wavefunction of all electrons = product of one-particle eigenfunctions. iii) one-particle eigenfunctions  atomic orbitals of H-like atoms. attainable by HF-SCF process. b) Slater’s approximation: screen constant & effective nuclear charge.

4 c) Addition of Angular momenta and its physical meaning/applications in understanding electronic states of many-e atom. i) Details of the e-e interactions can be symbolized by L (ML) and S(MS)! ii) Atomic spectral terms: physical meaning (energy state of an electronic configurations that contain one to several microstates). derivation of terms for a given electronic configuration (normally simple) containing equivalent electrons and/or nonequivalent electrons. Pauli exclusion and Hund’s rules, and derivation of the ground-state term for a given electronic configuration.

5 2.7 计算Li2+的 所描述状态的能量E、角动量L2的平均值
要点:1)类氢离子能量及角动量本征方程表达式;2)本征函数正交归一性;3)态叠加原理;4)求平均值方法。 解: nlm均为Li2+原子轨道波函数(本征函数), 1 和2 均为多个nlm的线性组合,依态叠加原理,表示了Li2+的可能状态。先判断它们是否归一化: 即已归一化。(利用本征函数的正交归一性质)

6 类氢离子的 等原子轨道波函数均正交归一,且主量子数n相同,能级简并,其能量均为: 故两个波函数所描述状态的能量可由平均值公式 分别导出: 又原子轨道的L2值由本征方程 确定为: 则两个波函数的L2值由求平均值方法分别导出:

7 2.25 已知N原子的电子组态为1s22s22p3 (1) 叙述其电子云分布特点; (2) 写出N的基态光谱项与光谱支项; (3) 写出激发态2p23s1的全部光谱项。 解:1)电子云呈球状分布: 1s、2s轨道上电子云本身就是球状分 布;2p亚层轨道半满填充,电子云分布接近球状。 2) 基态电子组态2p3,由洪特规则其最优电子排布为:  (MS)max =3/2, 即 S=3/2,此时 L=1+0-1=0, J =|L-S| = 3/2, 故基态光谱项为4S, 光谱支项为 4S3/2.

8 3) 先将相互作用弱的亚层分开处理,即有: 2p2 组态谱项: 3P(L=1, S=1), 1D(L=2,S=0), 1S(L=0,S=0); 3s1组态谱项: 2S (L=0, S=1/2) 则根据角动量加和原则可得: 3P  2S = 4P, 2P; 1D  2S = 2D; 1S  2S = 2S 故2p23s1组态的光谱项有4P、2P、 2D、2S。

9 原子第一电离能近似于最高占据轨道能量的负数,以下为H-Xe原子的第一电离能分布图,由近似公式可以看出 这些IP变化反映了有效核电荷,请根据有效核电荷和屏蔽常数讨论IP数值变化规律。
要点: 1) slater规则与屏蔽常数、有效核电荷; 2) slater近似下轨道能级表达式。

10 He: 1s = 0.3 Zeff = 2-0.3 =1.7 Zeff/n = 1.7
Li: 2s = 2x Zeff = = Zeff/n = 0.65 Be: 2s = 2x Zeff = = Zeff/n = 0.975 B: 2p = 2x0.85+2x0.35 Zeff = = Zeff/n = 1.3 C: 2p = 2x0.85+3x0.35 Zeff = = Zeff/n = 1.625 N: 2p = 2x0.85+4x0.35 Zeff = = Zeff/n = 1.95 O: 2p = 2x0.85+5x0.35 Zeff = = Zeff/n = 2.28 F: 2p = 2x0.85+6x0.35 Zeff = = Zeff/n = 2.60 ……. Mg:3s = 2x1.0+8x Zeff = Zeff/n = 0.9 Ca: 4s = 10x1.0+8x Zeff = Zeff/n = 0.71 Sr: 5s = 28x1.0+8x Zeff = Zeff/n = 0.57 同一周期:由左至右,最高占据轨道能量依序降低,IP 依序升高。 同一主族:由上至下,最高占据轨道能量依序升高,IP 依序降低。

11 思考题: 氢原子或类氢离子中核外电子围绕原子核运动,其运动状态由波函数(原子轨道)来描述,量子态的能量由主量子数n确定,表示为
试由此估算电子的动能、势能及平均半径。

12 提示: 可以根据第一章中的维里定理来推导 (V=-2T, E = V+T),也可以直接写出动能和势能的表达式并利用电子围绕原子核运动时离心力等于向心力(即核-电子静电吸引力)来推导。
电子围绕原子核运动的离心力等于核和电子相互作用力:

13 提示: 可以根据第一章中的维里定理来推导 (V=-2T, E = V+T),也可以直接写出动能和势能的表达式并利用电子围绕原子核运动时离心力等于向心力(即核-电子静电吸引力)来推导。


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