《无机化学 Chemical Theory》课程教学资源（讲稿）Part II Molecular Spectroscopy Chapter 5 Molecular Orbitals 分子轨道

Part I11Symmetry and BondingChapter 5MolecularOrbitals（分子轨道）Prof.Dr.XinLu（吕鑫）Email: xinlu@xmu.edu.cnhttp://pcoss.xmu.edu.cn/xlv/index.htmlhttp://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html

Part III Symmetry and Bonding Chapter 5 Molecular Orbitals（分子轨道） Prof. Dr. Xin Lu （吕鑫） Email: xinlu@xmu.edu.cn http://pcoss.xmu.edu.cn/xlv/index.html http://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html

Reviewingdirect products·如果两个函数分别按不可约表示r）和rG变换，那么他们的乘积函数当按这两个不可约表示的直积r?r变换·两个不可约表示直积中各对称操作的特征标就是两个不可约表示相应特征标的乘积每个对称操作的特征标的乘积：(a,b,c,··）?(p,q,r,.)=(a×p,b×q,c×r,.·)·每个点群必有一个全对称不可约表示rtot.sym.-所有操作的特征标均为+1·任一不可约表示r(i)与全对称不可约表示的直积就是该表示本身：r(i)?rtot.sym.=『r(i)·任意一维不可约表示和它自身的直积就是全对称不可约表示：『(i)?『(i)=『tot.sym·任意高维不可约表示和它自身的直积r(i)?r(i）必包含全对称不可约表示rtot.sym。·标量（数字）（numbers）按全对称不可约表示变换

Reviewing—direct products • 如果两个函数分别按不可约表示 Γ (i) 和 Γ (j)变换, 那么他们的乘积函数当按这两个不 可约表示的直积Γ (i)⊗ Γ (j)变换. • 两个不可约表示直积中各对称操作的特征标就是两个不可约表示相应特征标的乘积 每个对称操作的特征标的乘积: (a, b, c, . . .) ⊗ (p, q, r, . . .) = (a×p, b×q, c×r, . . .) • 每个点群必有一个全对称不可约表示Γ tot. sym. - 所有操作的特征标均为 +1。 • 任一不可约表示Γ (i)与全对称不可约表示的直积就是该表示本身: Γ (i)⊗ Γ tot. sym.= Γ (i) . • 任意一维不可约表示和它自身的直积就是全对称不可约表示： Γ (i)⊗ Γ (i)= Γ tot. sym. • 任意高维不可约表示和它自身的直积Γ (i)⊗Γ (i )必包含全对称不可约表示Γ tot. sym. 。 • 标量(数字) (numbers) 按全对称不可约表示变换

Reviewingvanishing integrals1．若函数山不按全对称不可约表示变换，则其积分I=「山dt必为零。2．若两个原子的AO波函数，和山，不依同一不可约表示变换，则其重叠积分Sij=『,山,dt必为零。换句话说，对称性相同(依同一不可约表示变换)的原子轨道间才可以有效重叠。3.矩阵元Q=了,Q,dt的值必为零若对应的直积r?r(Qr)不含全对称不可约表示。4.哈密顿算符必然按全对称不可约表示变换(？!)，若两个原子的AO波函数Φ和山不依同一不可约表示变换，则交换积分βi=H山,dt 必为零。即对称性相同（依同一不可约表示变换的原子轨道间才可以有效成键，形成分子轨道

Reviewing—vanishing integrals 1. 若函数ψ不按全对称不可约表示变换，则其积分𝑰 = ψ𝒅𝝉 必为零 。 2. 若两个原子的AO波函数ψi 和 ψj不依同一不可约表示变换，则其重叠积分𝑺𝒊𝒋 = ψ𝒊 ∗ψ𝒋 dτ 必为零。换句话说，对称性相同(依同一不可约表示变换)的原子轨道间才 可以有效重叠。 3. 矩阵元𝑸𝒊𝒋 = ψ𝒊 ∗𝑸 ψ𝒋 dτ 的值必为零若对应的直积 Γ (i) ⊗ Γ (Q)⊗ Γ (j) 不含全对称不可约 表示。 4. 哈密顿算符必然按全对称不可约表示变换(? !)，若两个原子的AO波函数ψi 和 ψj 不依同一不可约表示变换，则交换积分 𝜷𝒊𝒋 = ψ𝒊 ∗𝑯 ψ𝒋 dτ 必为零。即对称性相同（依 同一不可约表示变换）的原子轨道间才可以有效成键，形成分子轨道

5.Molecular orbitals.Nowthatwehave developedthenecessaryGroupTheorytools,wecanusethemtodrawup（qualitative）MOdiagrams.（注：这是正则分子轨道（canonicalmolecularorbitalCMO）图像，而非大一时学过的定域分子轨道（LMO)图像！）Symmetry arguments greatly simplify this process and help us not only to work out whichinteractions are important butalsomake it possibleto sketch theform ofthe MOs inastraightforwardway· In addition, we will be able to say something about the resulting electronic properties ofthemolecule and discuss why molecules havea preferencefor one shapeoveranother

5. Molecular orbitals • Now that we have developed the necessary Group Theory tools, we can use them to draw up (qualitative) MO diagrams. （注：这是正则分子轨道（canonical molecular orbital, CMO）图像，而非大一时学过的定域分子轨道(LMO)图像！） • Symmetry arguments greatly simplify this process and help us not only to work out which interactions are important but also make it possible to sketch the form of the MOs in a straightforward way. • In addition, we will be able to say something about the resulting electronic properties of the molecule and discuss why molecules have a preference for one shape over another

5. Molecular orbitalsThe procedure we will adopt for drawing up MO diagrams :1.Identifyingthepointgroupof themoleculetobeconcerned2. Identifying the AOs (valence orbitals) to be involved in bonding3.Classifying theAOs according to symmetry and, if necessary, combining thosesymmetrically equivalent AOs to form symmetry orbitals, SOs4. Allowing orbitals of the same symmetry to overlap (both in phase and out of phase!), andhence constructing the MO diagram(In the Chapter of"Representations,we have learnt some concepts needed in step 3.)

5. Molecular orbitals The procedure we will adopt for drawing up MO diagrams： 1. Identifying the point group of the molecule to be concerned. 2. Identifying the AOs (valence orbitals) to be involved in bonding. 3. Classifying the AOs according to symmetry and, if necessary, combining those symmetrically equivalent AOs to form symmetry orbitals, SOs. 4. Allowing orbitals of the same symmetry to overlap (both in phase and out of phase!), and hence constructing the MO diagram. (In the Chapter of “Representations”, we have learnt some concepts needed in step 3.)

5.1 Basic observations about MOs. When two AOs of the same symmetry interact, a bonding MO is formed which is lower inenergy than the lowest energy AO and an antibonding MO is formed which is higher inenergy than the highest energy AO.out-of-phasegreatestcontributionoverlapfromAantibondingloweringinenergybondingin-phasesmalleroverlaploweringevensmallerloweringBNote:thesizeofanAOalsogreatestcontributionmattersinbonding!increasingenergyseparationbetweenAOsfromB

5.1 Basic observations about MOs • When two AOs of the same symmetry interact, a bonding MO is formed which is lower in energy than the lowest energy AO and an antibonding MO is formed which is higher in energy than the highest energy AO. Note: the size of an AO also matters in bonding!

一5.1 Basic observations about MOs.Whenseveral AOs interacttoformMOs,thenumber of theMOsis the sameasthenumberof the AOshigherthanhighestAOdetailed calcneededtofindenergieslowerthanAOsMOsAOsMOsAOsMOslowestAO. In this more complex case it remains true that a particular MO will have the greatestcontribution from the AOs which are closest to it in energy

5.1 Basic observations about MOs • When several AOs interact to form MOs, the number of the MOs is the same as the number of the AOs. • In this more complex case it remains true that a particular MO will have the greatest contribution from the AOs which are closest to it in energy

Representing MOs: To draw MOs, we need to show the result of the in-phase or out-of-phase overlap, as wellastherelativecontributionsmadebythedifferentAOsOut-of-phase overlapgreatestcontributionfromatomAequal contributionIn-phase overlapgreatestcontributionfromatomB(white~positive,black~negative)

Representing MOs • To draw MOs, we need to show the result of the in-phase or out-of-phase overlap, as well as the relative contributions made by the different AOs. (white ~ positive, black ~ negative) In-phase overlap greatest contribution from atom A. greatest contribution from atom B. Out-of-phase overlap equal contribution

5.2 MO diagram for watero-x2OJEC2vC2: Example: H,O (point group C2v)1x2;y2;22111AlZ:TheO ls AOis too contracted and too low in11-1A2-1Rzxyenergy, transforming as A,B-11-1R,1xXzB21-1-11Rxyyz·0: 2s(spherical) as Aj:0202(Sa, St)T=A, @B2pz(z-like) as Ar.By inspection! (ForAOs without equivalentAOs)2px(x-like) as Bi2py(y-like) as B,. 2H: (sa ss) A, B, (Already considered in chapter 2)as A,0, = (sa+ sb)(x-like) as B)0,=(Sa-St)

5.2 MO diagram for water • Example: H2O (point group C2v) • The O 1s AO is too contracted and too low in energy, transforming as A1 . • O: 2s 2pz 2px 2py • 2H: (sa , sb ) 1 = (sa+ sb ) as A1 2= (sa -sb ) (x-like) as B1 (Already considered in chapter 2) (spherical) as A1 ; (z-like) as A1 . (x-like) as B1 (y-like) as B2 By inspection! (For AOs without equivalent AOs) A1  B1 (sa , sb ) 2 0 2 0  = A1  B1 b a

5.2 MO diagram for water: A rough sense of the relative energies of the AOs involved is needed to draw up the MOdiagram0O2sAO<2pAOD322名SS&SiHD0CHeIBePAr· H 1s AOs ~the oxygen 2p AOs23151840-203.5-40Ae/Koioue-60-80-100-120S-140

5.2 MO diagram for water • A rough sense of the relative energies of the AOs involved is needed to draw up the MO diagram. • O 2s AO < 2p AO, • H 1s AOs ~ the oxygen 2p AOs