/University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009

Size: px

Start display at page:

Download "/University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009"

Colin Shelton
5 years ago
Views:

1 Lecture 0 /6/09 /Univerity of Wahington Department of Chemitry Chemitry 453 Winter Quarter 009. Wave Function and Molecule Can quantum mechanic explain the tructure of molecule by determining wave function and energie for electron in molecule? major challenge i the fact that Schoedinger equation i quite complicated for molecule compoed of multiple nuclei and multiple electron. Conider Schroedinger equation for the hydrogen molecule. We have four kinetic energy term correponding to the motion of proton and two electron and we have ix term in the potential energy which reflect the ix Coulombic interaction. Uing the horthand notation:, the Hamiltonian for the hydrogen molecule i x y (ee diagram below, left) h h e F H c h c h 8π M 8π m HG 0 The firt four term are the kinetic energie of the proton ( and ) and the two electron ( and ). ecaue the ma of the proton i much greater than the ma of the electron (M>>m), the kinetic energie of the proton can be neglected. Thi i called the orn-oppenheimer pproximation and yield the impler approximate Hamiltonian: h e F H c h m HG KJ 8π 0 Even with thi implified Hamiltonian..the Schroedinger equation HΨEΨ cannot be olved exactly. KJ

2 Hydrogen Molecule H Hydrogen Molecule on H. Variational Principle and the LCO Method f an exact wave function cannot be obtained by directly olving Schroedinger equation, approximate wave function and energie can be obtained uing the Variational Principle. The Quantum Mechanical Variational Principle tate: Given any approximation wave function atifying the boundary condition of the problem, the expectation value of the energy calculated with thi wave function will alway be higher than the true energy of the round tate of the ytem. Thi principle provide a method for obtaining approximate wave function and energie in ituation where the Schroedinger equation cannot be olved Example: The implet molecule i the hydrogen molecule ion H, which ha only a ingle electron. The Hamiltonian i (ee diagram above, right) h e F H, where the orn-oppenheimer 8π m 0HG KJ pproximation ha again been ued. To obtain the expectation value or the average value of the energy, multiply the Schroedinger equation on the left by the complex conjugate of the wave function: Ψ* H Ψ Ψ* E Ψ E Ψ* Ψ. Note the wave function Ψ* cannot be hifted to the right of the Hamiltonian becaue the Hamiltonian operate on anything to it right ntegrate both ide of the Schroedinger equation over all pace Ψ* HΨd Ψ* EΨd E Ψ* Ψd where ddx dy d. Solve for E which i the expectation value E E * H Ψ Ψ d Ψ* Ψd We could olve for the energy if we knew the wave function. ut the bet we cannot do i gue a probable form for the wave function. We gue that the wave function i

3 a linear combination of atomic orbital (LCO), pecifically the orbital: Ψ c c, where c and c are contant to be determined and the wave function are the wave function for hydrogen atom and. Subtituting the trial wave function into the energy equation we ch cch ch obtain E c ccs c H H d H H d H H d H d S d d The Variational Principle tate that the energy calculated uing the trial wave function Ψ c c will be higher than the true ground tate energy. Therefore, the true value of c and c can be obtained by minimiing the energy of the ytem with repect to thee contant. Thi can be done uing method that are tandard in the calculu. C. Variational Principle: onding and nti-bonding Orbital pplying the variational principle, the energy of the electron in H can be obtained by minimiing the variational energy with repect to the contant c and c We obtain two ecular equation: E 0 cbh Eg cbh S Eg c E 0 cbh SEg cbh Eg c Solving thee two equation for the energy we get two olution E H ± H ± ± S where we have ued the fact that H H. Subtituting each of thee energie back into the ecular equation we obtain two olution c ±c. Thi yield two orbital each with a different energy For E c c c Ψ c h E Ψ c c c * ± ± We require that both orbital be normalied: ΨΨ from which we obtain c S c h c h, d c ± d Ψ± ± S c h

4 c onding Orbital: Ψ S h i called the bonding orbital and i H H given the horthand σ. The bonding orbital σ a an energy E S The Coulomb integral H ha the form e F H E d e d E d HG F KJ HG KJ The firt to term E d d, are both le than 0 bg 0 ero They reflect the tabiliation of the molecule a a reult of the attraction of the electron for the econd nucleu, and would be expected to reult in a more table orbital energy than the iolated hydrogen atom. ut a the atom approach the nuclear repulion term > 0 grow larger and hence the net 0 contribution of H to tabiliation i mall. The tabiliation of the H ion derive from the reonance integral: e F H H d G c h c h* * 0 d c h * dj c h H c h e * * H0 d d 0 0 F H G Phyically thi integral i the tabiliation deriving from the fact that the electron can hift from the orbital to the orbital. t i called the reonance integral becaue of the reemblance to the coupling or reonance of two claical ocillator. a reult of the negative reonance integral, the energy of an electron in the bonding orbital of H i le than the energy of the electron in the atomic orbital of a hydrogen atom. plot of E, the energy of the σ molecular bonding orbital a a function of the internuclear ditance i hown below, right. The nti-bonding Orbital: Ψ S h i called the anti-bonding orbital and i given the horthand σ. The bonding orbital ha an energy H H E. Thi energy will be higher than the energy of an iolated S hydrogen atom. Thi mean that if the electron were in the σ orbital of H, the molecule would be le table than the eparated hydrogen atom and ion. c S K J K

5 . Molecular Orbital for Diatomic Molecule Molecular orbital can be formed from the combination of other atomic orbital. The combination of orbital to form bonding and anti-bonding orbital can be repeated with orbital. The reult i analogou. The molecular orbital wave function have the form Ψ ± c ± h, where S d S tomic p orbital can be combined into linear combination to form molecular orbital but the ituation i complicated by the fact that p orbital are dependent on θ and φ, reulting in two general molecular orbital: Sigma (σ p ) orbital: When p orbital are combined with their lobe along the bonding axi (i.e. the axi drawn through the two nuclear coordinate) the reult i a igma orbital, which i axially ymmetric about the bonding axi there i alo a correponding anti-bonding σ p orbital Sigma orbital derived from linear combination of orbital:

6 Sigma orbital derived from linear combination of p orbital: Ψ σ p px px Pi (π p ) orbital: When p orbital are combined with the bonding axi entirely within the nodal plane, the reult i a molecular orbital which doe not have axial ymmetry about the bonding axi. Such orbital are called π orbital. There are two mutually orthogonal et of π orbital in diatomic molecule, deignated π y and π. They have equivalent energie (i.e. degenerate). ociated with thee orbital are antibonding orbital, deignated π y and π. The anti-bonding orbital alo have equivalent energie.

7 n diatomic molecule, electron fill molecular orbital in the order of increaing y y energy: σ σ < σ < σ < σ < π π < π < π < < p p p p p σ p Two electron can occupy each molecular orbital, but thee electron mut have oppoite pin angular momentum which i the angular momentum aociated with the rotational motion of the electron charge a oppoed to the angular momentum aociated with the orbital motion of the electron. The bonding order of a diatomic molecule i defined a the number of electron in bonding orbital minu the number of electron in anti-bonding orbital, divided by two. For a diatomic molecule to be table, the bonding order mut be greater than 0. f the bonding order i ero, the molecule i untable. Example: the bonding order of H, a table molecule, i. The bonding order of He i ero. He doe not exit. Example: O ha 6 electron. There are two electron in each of the MO orbital from σ to π p. The π y * and π * anti-bonding orbital have one electron each. The unpaired electron in the anti-bonding orbital impart a property to O called paramagnetim End of Lecture 3

Bogoliubov Transformation in Classical Mechanics

Bogoliubov Transformation in Classical Mechanics Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How