PCh10 Lec 3 - Revision history

Bes: /* Calculation of Energy */

2020-04-29T14:37:25Z

Calculation of Energy

Bes: /* Wavefunctions */

2020-04-29T14:36:35Z

Wavefunctions

Bes: /* Wavefunctions */

2020-04-29T14:35:33Z

Wavefunctions

Bes: /* Wavefunctions */

2020-04-29T14:34:12Z

Wavefunctions

Bes: /* Wavefunctions */

2020-04-29T14:32:02Z

Wavefunctions

Bes: /* Wavefunctions */

2020-04-29T14:25:32Z

Wavefunctions

Bes: /* Wavefunctions */

2020-04-29T14:19:47Z

Wavefunctions

Bes: /* Hamiltonian */

2020-04-29T14:00:46Z

Hamiltonian

Bes: /* Hamiltonian */

2020-04-29T13:58:52Z

Hamiltonian

Bes: /* The Big Picture */

2020-04-29T13:53:51Z

The Big Picture

← Older revision		Revision as of 14:37, 29 April 2020
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	:The above method is referred to as '''''self-consistent field method using the variational method,''''' and if you used a Hamiltonian approximation of Hartree-Fock, then it is called the H-F self consistent field method using the variational method.		:The above method is referred to as '''''self-consistent field method using the variational method,''''' and if you used a Hamiltonian approximation of Hartree-Fock, then it is called the H-F self consistent field method using the variational method.
	:Note: you can use the "self-consistent" criteria with other "methods" of changing/varying the trial wavefunction...and come up with all sorts of different "names" for calculation techniques. By the way, one reason i like drones and PID controllers is because they are applying a "self-consistent" strategy to maintain or change the position of the drone...i will control myself and not go off on a tangent.		:Note: you can use the "self-consistent" criteria with other "methods" of changing/varying the trial wavefunction...and come up with all sorts of different "names" for calculation techniques. By the way, one reason i like drones and PID controllers is because they are applying a "self-consistent" strategy to maintain or change the position of the drone...i will control myself and not go off on a tangent.
		+
		+	'''END of Lecture'''

@@ Line 40: / Line 40: @@
 [[File:Screen Shot 2020-04-27 at 9.48.37 AM.png|500px]]
-Q: What is the best value for ''ζ''?
+Q: What is the best value for ''ζ'' ?
-A: (see below) - the value that gives us the lowest energy...
+A: (see below in Energy discussion) - the value that gives us the lowest energy...
 ====Calculation of Energy====

@@ Line 38: / Line 38: @@
 In the discussion below we discuss how to solve for the energy. In this conversation we discuss ways of varying the mathematical form of the wavefunction. This topic will be discussed in much more detail, but for now let me remind you of a wavefunction from last lecture related to the use of an effective nuclear charge (''ζ'') as opposed to the true nuclear charge (''Z''):
-[[File:Screen Shot 2020-04-27 at 9.48.37 AM.png|600px]]
+[[File:Screen Shot 2020-04-27 at 9.48.37 AM.png|500px]]
 ====Calculation of Energy====

@@ Line 36: / Line 36: @@
 There is a method to make sure the wavefunction is antisymmetric that involves the use of "Slater [https://en.wikipedia.org/wiki/Determinant Determinant]." I am not going to go into details  here of how to generate antisymmetric wavefunction, but I would like for you to be able to solve a 2x2 and 3x3 determinant...this is a basic math operation. Visit the [https://en.wikipedia.org/wiki/Determinant determinant] wikipedia page for basics and work Example problem 10.1 in your notes.
-In the discussion below we discuss how to solve for the energy. In this conversation we discuss ways of varying the mathematical form of the wavefunction. This topic will be discussed in much more detail, but for now let me remind you of a wavefunction from last lecture
+In the discussion below we discuss how to solve for the energy. In this conversation we discuss ways of varying the mathematical form of the wavefunction. This topic will be discussed in much more detail, but for now let me remind you of a wavefunction from last lecture related to the use of an effective nuclear charge (''ζ'') as opposed to the true nuclear charge (''Z''):
-[[File:Screen Shot 2020-04-27 at 10.05.38 AM.png|400px]]
+[[File:Screen Shot 2020-04-27 at 9.48.37 AM.png|600px]]
 ====Calculation of Energy====

@@ Line 34: / Line 34: @@
 One aspect of an electron wavefunction, that is to be used in a multi-electron system is that we cannot tell the electrons apart...ie, we cannot label electrons with subscripts 1, 2, etc. Another way with saying this is that electrons are '''indistinguishable'''. This leads to ''Postulate 6'' in QM that requires all electron wavefunctions are '''antisymmetric''' as opposed to symmetric. If a 2-electron wavefunction is written as ''ψ(1,2)'' to indicate electron 1 and electron 2, then the wavefunction ''ψ(2,1)'' must also be an equivalent since the physical meaning of the wavefunction is ''ψ<sup>2</sup>''. If the wavefunction is antisymmetric then ψ(1,2)= - ψ(2,1), but ''ψ<sup>2</sup>'' are the same.
-There is a method to make sure the wavefunction is antisymmetric that involves the use of "Slater [https://en.wikipedia.org/wiki/Determinant Determinant]." I am not going to go into details  here of how to generate antisymmetric wavefunction, but I woul dlike for you to be able to solve a 2x2 and 3x3 determinant...this is a basic math operation:
+In the discussion below we discuss how to solve for the energy. In this conversation we discuss ways of varying the mathematical form of the wavefunction. This topic will be discussed in much more detail, but for now let me remind you of a wavefunction from last lecture
 ====Calculation of Energy====

← Older revision		Revision as of 14:19, 29 April 2020
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	====Wavefunctions====		====Wavefunctions====
		+	''Sec 10.3: Wavefunctions Must Reflect Indistinguishability of Electrons.''

		+	One aspect of an electron wavefunction, that is to be used in a multi-electron system is that we cannot tell the electrons apart...ie, we cannot label electrons with subscripts 1, 2, etc. Another way with saying this is that electrons are '''indistinguishable'''. This leads to ''Postulate 6'' in QM that requires all electron wavefunctions are '''antisymmetric''' as opposed to symmetric.

		+	[[File:Screen Shot 2020-04-29 at 9.18.53 AM.png\|300px]]
		+
		+	There is a method to make sure the wavefunction is antisymmetric that involves the use of "Slater [https://en.wikipedia.org/wiki/Determinant Determinant]." I am not going to go into details here of how to generate antisymmetric wavefunction, but I woul dlike for you to be able to solve a 2x2 and 3x3 determinant...this is a basic math operation:

	====Calculation of Energy====		====Calculation of Energy====

@@ Line 25: / Line 25: @@
 ''Sec 15.3: Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrodinger Equation.''
-As we have previously done, the Hamiltonian for a multi-electron system is relatively easy to write out, but toe solve it is not possible. So what part of the Hamiltonian is the tricky part...the electron-electron repulsion term. So let's just cancel this term from the Hamiltonian...and now call this Schrodinger equation the Hartree-Fock equation...done.
+As we have previously done, the Hamiltonian for a multi-electron system is relatively easy to write out, but it is not possible to solve. So what part of the Hamiltonian is the tricky part...the electron-electron repulsion term. So let's just cancel this term from the Hamiltonian...and now call this Schrodinger equation the '''''Hartree-Fock''''' equation.
 ====Wavefunctions====

← Older revision		Revision as of 13:58, 29 April 2020
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	====Hamiltonian====		====Hamiltonian====

		+	''Sec 15.3: Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrodinger Equation.''

		+	As we have previously done, the Hamiltonian for a multi-electron system is relatively easy to write out, but toe solve it is not possible. So what part of the Hamiltonian is the tricky part...the electron-electron repulsion term. So let's just cancel this term from the Hamiltonian...and now call this Schrodinger equation the Hartree-Fock equation...done.

	====Wavefunctions====		====Wavefunctions====

@@ Line 11: / Line 11: @@
 :''YES...sorry this is considered the easier, ''time-independent'' Schrodinger equation...the ''time-dependent'' Schrodinger equation is used in spectroscopy since there is a time-dependence of when you excite the system...it apply a pulse of RF in FT-NMR...or any light (UV, vis, IR, etc). Time-dependent quantum or "Quantum: Spectroscopy" is a graduate level class that most pchem PhD students will take.''
 <div align="center">
-quote by [https://en.wikiquote.org/wiki/Paul_Dirac Paul Dirac]...
+'''''A quote by [https://en.wikiquote.org/wiki/Paul_Dirac Paul Dirac]...'''''
-:'''''The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.'''''
+:'''The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.'''
 </div>