《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）ESM-3-Hatom

QMpostulates: y can describe all the properties of a system- though it is not a physical observable, yet lw2 is !- The form of , in = Hb;if V(r)only, H = Eat: Operator is linear (Operator·(a + p) =Operator-a +Operator-b follows the association law orrespects superposition, implies linearly expandability orlinear transformation of a function) and Hermitian(guarantees eigenvalue to be real, which corresponds tophysical observable).- Eigenfunctions of an operatorform a complete andorthonormal (linearly independent) setJidr = Sij and l ><il = I or f = cii2

QM postulates • ψ can describe all the properties of a system – though it is not a physical observable, yet |ψ| 2 is ! – The form of ψ, 𝑖ℏ 𝜕𝜓 𝜕𝑡 = ℋ𝜓; 𝑖𝑓 𝑉 𝑟 𝑜𝑛𝑙𝑦, ℋ𝜓 = 𝐸𝜓 • Operator is linear (Operator•(ψ𝑎 + ψ𝑏) = Operator•ψ𝑎 +Operator•ψ𝑏, follows the association law or respects superposition, implies linearly expandability or linear transformation of a function) and Hermitian (guarantees eigenvalue to be real, which corresponds to physical observable). – Eigenfunctions of an operator form a complete and orthonormal (linearly independent) set, ׬ ψ𝑖ψ𝑖𝑑𝑟 = 𝛿𝑖𝑗, 𝑎𝑛𝑑 σ𝑖=1 ∞ ψ𝑖 >< ψ𝑖 = I or 𝑓 = σ𝑖=1 ∞ 𝑐𝑖ψ𝑖 2

QMpostulates: Operator when acting upon any function producesonly its eigenvalues;- Operator A· f = Operator· Z=1 Cii ;one measurement of observable of operator = axc;and f collapses into x;- = J f*·OperatorfdrOperator A * Operator B ± Operator B * Operator Aobservables of non-commuting operators showuncertainty principle (no measurement involved);-Conjugate variable pairs obey Noether's Theorem,symmetry of one (variable) requires theconservation of the other (conjugate momentum)3

QM postulates • Operator when acting upon any function produces only its eigenvalues; – Operator A• f = Operator• σ𝑖=1 ∞ 𝑐𝑖ψ𝑖 ; one measurement of observable of operator = ax 𝑐𝑥 2 ; and f collapses into ψ𝑥; – = ׬�𝑑𝑓�•�𝑜𝑡𝑎𝑟𝑒𝑝𝑂�•∗�� • Operator A * Operator B ≠ Operator B ∗ Operator A, observables of non-commuting operators show uncertainty principle (no measurement involved); – Conjugate variable pairs obey Noether's Theorem, symmetry of one (variable) requires the conservation of the other (conjugate momentum) 3

NBOA version of valence bond theory (by Weinhold group, briefIntroduction)“"Natural Bond Orbitals(NBOs)are localizedfew-centerorbitalsthatdescribe the Lewis-like molecularbonding pattern of electronpairsin optimally compact form.More precisely,NBOs are anorthonormal set oflocalized"maximumoccupancy"orbitalswhoseleadingN/2 members give the most accuratepossibleLewis-likedescription of the total N-electron density";Occupancy of an orbital is not integer;Using natural atomic orbital (NAO) instead of H-like atomic orbital;Dependentonthemolecularenvironment(differentNAOsforthesame atoms in different molecules);ComparedtoMO,canbeviewedasbasis set (coordinate)switch,both describe the same total electron wavefunction ;4

NBO • A version of valence bond theory (by Weinhold group, brief Introduction) • “Natural Bond Orbitals (NBOs) are localized few-center orbitals that describe the Lewis-like molecular bonding pattern of electron pairs in optimally compact form. More precisely, NBOs are an orthonormal set of localized "maximum occupancy" orbitals whose leading N/2 members give the most accurate possible Lewis-like description of the total N-electron density”; • Occupancy of an orbital is not integer; • Using natural atomic orbital (NAO) instead of H-like atomic orbital; • Dependent on the molecular environment (different NAOs for the same atoms in different molecules); • Compared to MO, can be viewed as basis set (coordinate) switch, both describe the same total electron wavefunction ψ; 4

Singlepointenergy calculation summarySingle pointstructurewithknown atomic coordinates,orsingleconfiguration (stable or unstable structurebothOk)or a point on the potential energy surface;Single point energy calculation,·KeywordSP (omitted bydefault): Normal termination (SCF done);·Information about the electronic structure;? Energy and other property does not require structural change;.The only doable calculation;·Upon the optimized structureusing a lower level theory,calculations of a higher level theory can be carriedout tocompute some properties;·Energy of different model chemistry (e.g. HF/6-31G vs HF/3-21G)cannot be compared5

Single point energy calculation summary Single point structure with known atomic coordinates, or single configuration (stable or unstable structure both OK), or a point on the potential energy surface; Single point energy calculation, • Keyword SP (omitted by default) • Normal termination (SCF done); • Information about the electronic structure; • Energy and other property does not require structural change; • The only doable calculation; • Upon the optimized structure using a lower level theory, calculations of a higher level theory can be carried out to compute some properties; • Energy of different model chemistry (e.g. HF/6-31G vs HF/3-21G) cannot be compared 5

NoteonconstantsandunitsResource on physical constants:http://physics.nist.gov/cuu/Constants/Resource for unit conversions:http://www.digitaldutch.com/unitconverter/Atomicunitscommonforquantummechanicalcalculationshttp://en.wikipedia.org/wiki/Atomic_unitsSI unitAtomic unitCommon unit1.6021x10-19cChargee=15.29177x10-1l1mLength0.529177Ado= 1 (bohr)Mass9.10938x10-31kgme= 11.054572×10-34Jsh=1Angularmomentum4.359744×10-18 JEnergy27.2114eVEn (hartree)8.987552×10°c-2NmElectrostaticforce1/(4元80) = 11.38065x10-23JK-1Boltzmannconstant8.31447J/molK1eV= 1.60218X 10-19 J = 96.485 kJ/mol =8065.5 cm-1 = 11064K6

Note on constants and units 6 Resource on physical constants: http://physics.nist.gov/cuu/Constants/ Resource for unit conversions: http://www.digitaldutch.com/unitconverter/. Atomic units common for quantum mechanical calculations http://en.wikipedia.org/wiki/Atomic_units 1 eV = 1.60218×10−19 J = 96.485 kJ/mol = 8065.5 cm−1 = 11064 K

ContentsHydrogenatom1.Angular & radial component2.Variational principle3.Basisfunctions4.Secularequations5.Spin

Contents 7 Hydrogen atom 1. Angular & radial component 2. Variational principle 3. Basis functions 4. Secular equations 5. Spin

x=rsingcos1Hydrogenatomy=rsinesingz=rcosodsin drdd? Simplest chemical “thing"- Represented in spherical coordinates-Massive nucleus at origin00<x,Y.z<0<r<8-Mapmotionofelectron0<<0<2元-HatomSchrodingerequationv(r)+V(r)v(r)=Ev(r)2mee1Coulombpotentialdecaysslowlywithdistance.V(r):4元.re282r281ab1aonb1sineEu2μ2OrOrr2 sin0 00004元E0r2 sin? 0g2Separationofvariables-andpdon'tappearanywhereotherthantheLaplacianthreedegreeoffreedom=threequantumnumbery(r)=Ym (.0)R. (r)8

Hydrogen atom • Simplest chemical “thing” – Represented in spherical coordinates – Massive nucleus at origin* – Map motion of electron – H atom Schrödinger equation 8 Coulomb potential decays slowly with distance. Separation of variables – θ and φ don’t appear anywhere other than the Laplacian three degree of freedom = three quantum number

Hydrogenatom:angularcomponenty(r)=Ym (e,o)R, (r)Angular component- Yim are so-called “spherical harmonics", describe angularmotion/angularmomentum of electronAngularquantumnumbers- azimuthal / = O, 1, 2, ... "shape" of angular distribution magnetic m, =-l, -/+1, ., /"orientation"1=0.m=0."s"function1=1,m=0.±1,“Pz,Px,Py"functionssign change on opposite lobesof p

Hydrogen atom: angular component 9 • Angular component – Ylm are so-called “spherical harmonics”, describe angular motion/angular momentum of electron • Angular quantum numbers – azimuthal l = 0, 1, 2, . “shape” of angular distribution – magnetic ml = -l, -l+1, ., l “orientation” sign change on opposite lobes of p

Hydrogenatom:radialcomponente2h? d?11(1 + 1)rR(r)=ErR(r2m。dr24元r2m.r?Solve by ... looking up (analytical) answer!-13.6eVn =1, 2, 3, ..2a1nlonizationenergy of an H atom?1s → 2s energy?Thermalpopulationdistribution?10

Hydrogen atom: radial component 10 • Solve by . looking up (analytical) answer! • Ionization energy of an H atom? • 1s → 2s energy? • Thermal population distribution?

Hydrogenatom:radialcomponent21t(a)R10 (r)=2a,%e-r/or/2aR20 (r)= 2-1/2 a%(1-Rn = polynomial(rn-l) * exp(-r/n)4元0h?=0.529x10-10ma=me20.7: Polynomial part gives radial nodes0.6O0.5· Exponential decay form called a(0.4以Slaterfunction0.30.20,10.234rlaFa2sorbitalaporbital11Lowe,Table4-2

• Polynomial part gives radial nodes • Exponential decay form called a Slater function Hydrogen atom: radial component 11 Lowe, Table 4-2