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# Molecular Modeling CHEM 6970

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This 4 page Class Notes was uploaded by Mr. Clementine Gottlieb on Wednesday October 21, 2015. The Class Notes belongs to CHEM 6970 at Tennessee Tech University taught by Titus Albu in Fall. Since its upload, it has received 23 views. For similar materials see /class/225692/chem-6970-tennessee-tech-university in Chemistry at Tennessee Tech University.

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Date Created: 10/21/15

CHEM 6970 Fall 2004 BriefReview onuantum Mechanics Origins of Quantum Mechanics 7 The classical physics predicts the precise trajectory of a particle and allows the translational rotational and vibrational modes of motion to be excited to any energy by controlling the applied force These conclusions agree to everyday experience but do not extend to individual atoms Classical mechanics fails when applies to transfers of very small quantities of energy and to objects of very small mass Some early experimental evidence showedthat several concepts of classical mechanics are invalid 7 In the beginning of the 1920 s it was well established that the light behaves as a wave in some experiments and as a stream of photons in others This was known as the Wavepar cle duality of the light In 192324 Louis de Broglie proposed the idea that the matter might also display wavelike properties under certain conditions Under those conditions similar equations should hold for the matter A particle of mass m moving with the speed v will exhibit a de Broglie wavelength given by A g g 7 Schrodinger equation is the equation for nding the wave function of a particle and come up based on idea that if the matter possesses wavelike properties there must be a wave equation that governs them VWMW A Waufundinn 7 Wave nature ofthe particles 7 The wavefunction or state function of a system evolves in time according to the time a PUJ 71Enlh at 39 dependent Schrodinger equation Iflk xj ih where Pxt Vxe 7 The solutions of timeindependent Schrodinger equation are called stationary state wave functions i Woo 72m 1X2 Vxtx7Etx 7 Rewrite the equation and generalize to three dimensions hi 62 62 62 j 2 2 2 wxyzVxyzvxyzEvXJJ 239quot 8x 8y 82 CHEM 6970 Fall 2004 7 Rewrite including the notation for the Laplacian operator 62 62 a 2 2 2 V2 Laplacian operator 6x By 62 h2 2 3 V Vxyz lIEl 2m 7 Rewrite including the notation for the Hamiltonian operator 2 2 V2 Vx y z H Hamiltonian operator m 3 ill1 E 1 Simple form of the Schrodinger equation 7 By solving the Schrodinger equation for the hydrogen atom one obtains the allowed energies eigenvalues and the wave function of the electron eigenfunctions which will supply the atomic orbitals 7 Schrodinger equation in terms of the spherical coordinates 2 2 h L2326 h 2 1 15in96i 2me r Br 6r 8 2me r 51116 r h2 l 6211 82 2m 47rgrlEW e r s1n m9 0 7 Solutions are given as zr t9 RrYt9 and Yt9 6 12 l 1 39 32 Rn r 7 2 J rle7rna0 131431 lZnKn l3l quot00 na 0 Rquot r depends on two quantum numbers 2r Lilfl1 are associated Laguerre polynom1als nao 12 m 211llmlgt lml W Y 6 4 HM P cost9e lml I I P x are called associated dx m Pquotquot39x1 x27 Legendre polynomials CHEM 6970 Fall 2004 The Bomroppenhelmer approxlm at on 7 allow separatlon of eleetronle and nuelearmotlon Wmdleenlnr vR Weleetmnre n R lmlclzuR rHamlltoman for H2 moleeule t 2 2 2 2 v v i vfvg EE g 2M 2m2 475mm 47an LeL L i 39M r 3 l K 475D 47an 47an AnsDR H 24 7 Due to larger mass of the nuelel eompared wth the eleetrons thls appromma o onslders the nuelel flxedln posltlons relatlye to the mohon of the eleetrons negleetlng the nuelearmotlon Thls results ln droppmg two an lunehe energy terms ln the expresslon for the Hamlltonl 2 2 A i y y E 2 we 475mm 47an e The Hamlltonlan beeomes 2 2 2 2 2 eieiii 4mm Answzg 47an WEEK 7 The Hamlltonlan ln atomle unlts H 1VfV 1 i 2 R rThe ease of H a one eleetron system 7 Avolds the lntereleetronle repulslon term that makes the equahon not to be solved exaetly eThe Schrodlnger equatlon ean be solved exactly for H Wthln the Bom Oppenhelmer BO apprommatlon H looks more llke hydrogen atom oner W V m CHEM 6970 Fall 2004 Potential energy surfaces 7 An important property in discussing beam results and calculating reaction cross sections 7 According to the BomOppenheimer approximation the wavefunction of a system composed of N nuclei and n electrons is written as a product of nuclear wavefunction depending on the positions of the nuclei and an electronic wavefunction depending on the positions of the electrons within a xed nuclear con guration This allows solving the Schrodinger equation for the wavefunction of the electrons alone at a speci c nuclear con guration Modifying the nuclear con guration and solving again the Schrodinger equation one get a different energy Energy 7 The case of diatomic molecules representing the electronic energy versus the interatomic distance one obtains a potential energy curve 7 The case of triatomic molecules there are three geometric parameters that de ne the molecular geometry and representing all three of them plus the energy requires a 4dimensional representation 7 Example the geometry of water is de ned by bond lengths and one angle or A 3 bond lengths or 1 bond length and 2 bond angles or 3 bond angles HA 0 HB 7 The minimum number of geometric parameters necessary to de ne the geometry of a system ofN atoms N Z 3 is 3N 6 7 When the potential energy depends on more than one single geometric parameter then use the phrase potential energy surface 7 The entire potential energy function cannot be plotted because the plotting is limited to 3 dimensions To overcome this one usually represents the energy function of only 2 geometric parameters keeping the otherothers at a xed value Such a plot is a cross sectional cut of the full potential energy surface 7 Example water 7 a 3dimensional plot VrOHA rOHB a constant versus rOHA and rOHB gives information about how the potential energy of water molecule changes when the bond lengths are varied at a constant angle a A series of crosssectional plots at different values of a give information of how the potential energy depends on the angle a 7 The representation of the potential energy surface for the molecules with more than 3 atoms is much more complicated

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