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小木虫论坛-学术科研互动平台 » 计算模拟区 » 量子化学 » 量化新手直通车 » 【求助】电子态1Δg，1∑+g，2p等怎么定义的，
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zhang302

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[交流] 【求助】电子态1Δg，1∑+g，2p等怎么定义的，

我最近看文献，一直看到这个。实在有点不明白这些电子态是什么意思。只知道它是电子态。望大侠们给我解析一下，必有重金酬谢
其他电子态的说明也可以在下面留言，与前几个电子态一视同仁，金币不是问题
求讲解
越详细越好
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1楼 2011-03-31 10:17:39
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physics7778

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建议

★ ★
zhang302(金币+5): 提供了一个明确的查阅途径 2011-03-31 15:36:29
小红豆(金币+2): 谢谢分享你的看法 2011-04-16 14:06:28
引用回帖:
Originally posted by zhang302 at 2011-03-31 10:17:39:
我最近看文献，一直看到这个。实在有点不明白这些电子态是什么意思。只知道它是电子态。望大侠们给我解析一下，必有重金酬谢
其他电子态的说明也可以在下面留言，与前几个电子态一视同仁，金币不是问题
求讲解
...

建议你去看《分子光谱与分子结构》第一卷 双原子分子光谱 【加】G.赫兹堡  著
第六章  构造原理、电子组态和化学价。书上讲的很清楚。有需要可以传给你电子版的。
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2楼2011-03-31 11:39:19
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zhang302

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引用回帖:
Originally posted by physics7778 at 2011-03-31 11:39:19:
建议你去看《分子光谱与分子结构》第一卷 双原子分子光谱 【加】G.赫兹堡  著
第六章  构造原理、电子组态和化学价。书上讲的很清楚。有需要可以传给你电子版的。

如果能帮我贴出来更好，呵呵
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3楼2011-03-31 15:37:45
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yongma2008

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★ ★
zhang302(金币+10): 很感谢呀，我是不太会用度娘 2011-04-01 09:11:58
小红豆(金币+2): 谢谢分享你的看法 2011-04-16 14:06:39
引用回帖:
Originally posted by zhang302 at 2011-03-31 08:37:45:
如果能帮我贴出来更好，呵呵

其实度娘真的很有用，动动手啥都有了

http://good.gd/1081561.htm
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4楼2011-03-31 19:51:38
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jmlong

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★ ★ ★ ★ ★
zhang302(金币+10): 2011-04-01 09:10:58
小红豆(金币+5): 谢谢分享你的看法 2011-04-16 14:07:04
In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term symbol for the atomic case. However, the following presentation is restricted to the case of homonuclear diatomic molecules, or symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the u/g  symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian. In the case of less symmetric molecules the molecular term symbol contains the symbol of the group representation to which the molecular electronic state belongs.

It has the general form:


where

    * S is the total spin quantum number
    * Λ is the orbital angular momentum along the internuclear axis
    * Ω is the total angular momentum along the internuclear axis
    * u/g is the parity
    * +/− is the reflection symmetry along an arbitrary plane containing the internuclear axis

For atoms, we use S, L, J and MJ to characterize a given state. In linear molecules, however, the lack of spherical symmetry destroys the relationship [\hat\mathbf L^2, \hat H]=0, so L ceases to be a good quantum number. A new set of operators have to be used instead: \{\hat \mathbf S^2, \hat\mathbf{S}_z, \hat\mathbf{L}_z, \hat\mathbf{J}_z=\hat\mathbf{S}_z + \hat\mathbf{L}_z\}, where the z-axis is defined along the internuclear axis of the molecule. Since these operators commute with each other and with the Hamiltonian  on the limit of negligible spin-orbit coupling, their eigenvalues may be used to describe a molecule state through the quantum numbers S, MS, ML and MJ.

The cylindrical symmetry of a linear molecule ensures that positive and negative values of a given ml for an electron in a molecular orbital will be degenerate in the absence of spin-orbit coupling. Different molecular orbitals are classified with a new quantum number, λ, defined as

    λ = |ml|

Following the spectroscopic notation pattern, molecular orbitals are designated by a smallcase Greek letter: for λ = 0, 1, 2, 3,… orbitals are called σ, π, δ, φ… respectively.

Now, the total z-projection of L can be defined as

    M_L=\sum_i {m_l}_i.

As states with positive and negative values of ML are degenerate, we define

    Λ = |ML|,

and a capital Greek letter is used to refer to each value: Λ = 0, 1, 2, 3… are coded as Σ, Π, Δ, Φ… respectively. The molecular term symbol is then defined as

    2S+1Λ

and the number of electron degenerate states (under the absence of spin-orbit coupling) corresponding to this term symbol is given by:

    * (2S+1)×2 if Λ is not 0
    * (2S+1) if Λ is 0.

Ω and spin-orbit coupling

Spin-orbit coupling lifts the degeneracy of the electronic states. This is because the z-component of spin interacts with the z-component of the orbital angular momentum, generating a total electronic angular momentum along the molecule axis Jz. This is characterized by the MJ quantum number, where

    MJ = MS + ML.

Again, positive and negative values of MJ are degenerate, so the pairs (ML, MS) and (−ML, −MS) are degenerate: {(1, 1/2), (−1, −1/2)}, and {(1, −1/2), (−1, 1/2)} represent two different degenerate states. These pairs are grouped together with the quantum number Ω, which is defined as the sum of the pair of values (ML, MS) for which ML is positive. Sometimes the equation

    Ω = Λ + MS

is used (often Σ is used instead of MS). Note that although this gives correct values for Ω it could be misleading, as obtained values do not correspond to states indicated by a given pair of values (ML,MS). For example, a state with (−1, −1/2) would give an Ω value of Ω = |−1| + (−1/2) = −1/2, which is wrong. Choosing the pair of values with ML positive will give a Ω = 3/2 for that state.

With this, a level is given by

    2S + 1ΛΩ

Note that Ω can have negative values and subscripts r and i represent regular (normal) and inverted multiplets, respectively.[1] For a 4Π term there are four degenerate (ML, MS) pairs: {(1, 3/2), (−1, −3/2)}, {(1, 1/2), (−1, −1/2)}, {(1, −1/2), (−1, 1/2)}, {(1, −3/2), (−1, 3/2)}. These correspond to Ω values of 5/2, 3/2, 1/2 and −1/2, respectively. Approximating the spin-orbit Hamiltonian to first order perturbation theory, the energy level is given by

    E = A ML MS

where A is the spin-orbit constant. For 4Π the Ω values 5/2, 3/2, 1/2 and −1/2 correspond to energies of 3A/2, A/2, −A/2 and −3A/2. Despite of having the same magnitude, levels of Ω = ±1/2 have different energies associated, so they are not degenerate. With that convention, states with different energies are given different Ω values. For states with positive values of A (which are said to be regular), increasing values of Ω correspond to increasing values of energies; on the other hand, with A negative (said to be inverted) the energy order is reversed. Including higher order effects can lead to a spin-orbital levels or energy that do not even follow the increasing value of Ω.

When Λ = 0 there is no spin-orbit splitting to first order in perturbation theory, as the associated energy is zero. So for a given S, all of its MS values are degenerate. This degeneracy is lifted when spin-orbit interaction is treated to higher order in perturbation theory, but still states with same |MS| are degenerate in a non-rotating molecule. We can speak of a 5Σ2 substate, a 5Σ1 substate or a 5Σ0 substate. Except for the case Ω = 0, these substates have a degeneracy of 2.


Reflection through a plane containing the internuclear axis

There are an infinite number of planes containing the internuclear axis and hence there are an infinite number of possible reflexions. For any of these planes, molecular terms with Λ > 0 always have a state which is symmetric with respect to this reflection and one state that is antisymmetric. Rather than labelling those situations as, e.g., 2Π±, the ± is omitted.

For the Σ states, however, this two-fold degeneracy disappears, and all Σ states are either symmetric under any plane containing the internuclear axis, or antisymmetric. These two situations are labeled as Σ+ or Σ−.

Reflection through an inversion center: u and g symmetry

Taking the molecule center of mass as origin of coordinates, consider the change of all electrons’ position from (xi, yi, zi) to (−xi, −yi, −zi). If the resulting wave function is unchanged, it is said to be gerade (German for even); if the wave function changes sign then it is said to be ungerade (odd). For a molecule with a center of inversion, all orbitals will be symmetric or antisymmetric. The resulting wavefunction for the whole multielectron system will be gerade if an even number of electrons is in [[ungerade orbitals]], and ungerade if there is an odd number of electrons in ungerade orbitals, independently of the number of electrons in gerade orbitals.

[ Last edited by jmlong on 2011-4-1 at 06:09 ]
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5楼2011-04-01 06:01:35
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yongma2008

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引用回帖:
Originally posted by zhang302 at 2011-03-31 08:37:45:
如果能帮我贴出来更好，呵呵

不太会用度娘，我无语了，连基本搜索都不会？
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6楼2011-04-01 16:23:40
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zhang302

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澄清不会用度娘的原因

引用回帖:
Originally posted by yongma2008 at 2011-04-01 16:23:40:
不太会用度娘，我无语了，连基本搜索都不会？

我之前在网上也搜过这些东西。感觉特别的零碎，没有找到一个合理的答案。我是希望能借助虫友帮我详细的解答一下。那段时间，老师催任务催的特别的紧，就没下多大的功夫查找，多谢你的教训
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7楼2011-04-16 10:07:49
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jjww0001

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小红豆(金币+2): 谢谢分享你的看法 2011-04-16 14:07:22
不用这么复杂，看大众版的结构化学教材就行了，周公度的应该有，或者厦大林梦海的也有，实在不行先看懂原子光谱项就应该懂分子光谱项了
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8楼2011-04-16 13:35:18
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beefly

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小木虫(金币+0.5):给个红包，谢谢回帖
2p：2为主量子数，p为轨道量子数。在任何一本原子物理（以及原子光谱学、结构化学，量子化学）书里都能找到。

1Δg，1∑+g：

1为自旋多重度；

Δ，∑+仅用于线形分子（非线性分子需要查群表），表示轨道量子数在分子轴上的投影。例如d=2的投影，按照轨道磁量子数可以分为-2/+2，-1/+1，0三类，分别用Δ，Π，∑描述，显然除了∑以外都是二重简并的。至于∑+中的+号和∑-中的-号，分别表示0投影是对称的还是反对称的（也就是说，0有+0和-0两种）。在线性分子的轨道中，只能出现σ+（按照分子光谱学中的约定，这里用小写符号表示轨道，电子态用大写）；而在线性分子的电子态中，∑+和∑-都可以出现，但∑+个数往往更多。

g：正宇称对称性。相对应的还有u：反宇称对称性。在具有D∞h对称性的线性分子中会出现g/u对称性，例如同核双原子分子。原子是特殊情况：所有的s,d,g,...轨道（不是电子态）都是g对称性，所有的p,f,h,...都是u对称性。
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9楼2011-04-18 03:53:08
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