Hi sir. Have some questions regarding Hartree Fock. Is it possible to solve a multi-electron atom-orbital with the Roothan equations? By assuming that an AO is a linear combination of a basic set, or does it apply only to MO? The book I am using in my course (Peter Atkins Molecular quantum mechanics) says that Roothaan is used for MO orbitals, while pure HF is used for atoms. My problem with this is that I believe it will be very difficult to change the AOs between each iteration, as you sort of mentioned in your video. But if one was to use pure HF, what would one use as initial atomic orbitals? One final question about MO orbitals. When approximated as a linear combination of atomic orbitals, how does spin work? Does one multiply spin with the molecular orbital or is it fixed by itself when summarizing the linear combination of atomic orbitals (spin orbitals). I suppose you use spin molecule orbital in the Roothan equations, is this right?
Hi Sebastian. Let's address these questions one by one. 1). The word "orbital" inherently refers to one-electron wavefunctions. If we are solving for a wavefunction with multiple electrons we can use a Slater determinant of orbitals as an approximation to that multi-electron wavefunction, but it will be an approximation. That's what we're doing in HF-Roothaan, is minimizing the energy of a set of occupied orbitals. 2). A "molecular orbital" is an orbital in a molecule, whereas an "atomic orbital" is an orbital in an atom. Though when we say "atomic orbital" we often mean "atomic orbital basis function", which is used for the purposes of approximating a molecular orbital as a linear combination of these basis functions. 3). HF is used both for atoms and molecules. Any molecular system of more than one electron requires the use of approximate methods, and HF is one of the primary first approximations used to estimate the energy of both atomic and molecular systems. 4). As you mention, it is very difficult to solve the HF equations without some type of basis set. There are methods that can be used for very small systems, but which quickly become computationally intractible for sizeable systems. Essentially any method of numerically integrating a PDE can work in principal, though most common techniques are wildly inefficient. 5). Typically atomic HF uses AO basis functions, but only centered at a single nucleus, rather than at the multiple nuclei that would be in a molecule. The AO basis functions used for a carbon atom calculation typically wouldn't be any different than those used for a carbon atom within a molecular calculation. 6). Spin is an independent attribute of the spatial distribution of electrons, and is treated in the setup of the algorithm. My dedicated HF chapter in the Computational Chemistry playlist goes more in depth on the details of how spin is dealt with in the simplest HF flavor, restricted HF (RHF). 7). One can use spatial orbitals or spin orbitals in RHF, but spin is typically dealt with at the MO level, and is not attributed to AO basis functions.
So, there is a KxK F and S matrices for each individual atomic orbital, right? A comment, it should be easier to follow if you'd omit the subscript '1'. The derivation is not only for electron 1
Hi Miguel. The K by K matrix is for *all* atomic orbitals and molecular orbitals. The rows are coefficients for a particular atomic orbital, and the columns are coefficients for a particular molecular orbital (for the C matrix anyway).
F12 for example is an integral that depends on two different basis sets phi star 1 and phi 2. these are individual atomic orbital basis functions. he says that at 7:30 I'm not sure if he still answers questions to this day lol.
hi ,what i do exacly need is finding how to get the expression of each Fuv, Suv and coefficients c as well which u already menthioned them in your vedio u just have written their expression but how did they get to that i need their demonstration please sir
The actual values of the matrix elements are integrals whose formulas are described here, but not analytically computed. Such details are beyond the scope of this course, but are described thoroughly in "Molecular Electronic Structure Theory" by Helgaker ("the purple book" as it is often called). Some (but definitely not all) of these aspects will be surveyed in future Computational Chemistry videos. To compute an arbitrary Fock matrix element requires specifying a form of the atomic orbital basis functions (typically linear combinations of Gaussian functions with various angular components). For standard Gaussian-type orbitals, these formulas are reasonably straightforward if you're only using s orbitals, but they get obscenely complicated as you climb the angular momentum ladder (p, d, f, g, h, ...). At very high angular momenta they are often approximated rather than using the recursion equations. You could find these formulas in Helgaker, but the naive implementation scales very poorly, so I would only recommend using them for low angular momenta (s, p, d) and/or for small toy applications.
The purple equation is equivalent to the following yellow equation. It's the yellow matrix multiplication equation for an individual row / column. The purple equation is true for all integer values of mu such that 1
Thank you. it's still uncomfortable to accept. for me, the former one is a whole matrix and the latter one just a one of elements. may i think that they're conventionally treated as same?
Maybe it should be specified in the video that k different purple equations are generated each with a different value of mu (- [1,k]. P.S.:Very good videos by the way
Sure thing. At this level, resources start becoming very scarce, and web searches are often not helpful. Many of the best electronic resources are often some random professor's online notes, as many of these things aren't formally published in the literature, and you have to know somebody who knows what book to look into. I had a similar situation 2 years ago, and it took me 9 months until I was finally able to ask somebody who knew what book to look in, and once I knew I looked it up I had the solution coded and passing unit tests in an hour. Sometimes 30 seconds of good advice is worth weeks of solo struggle.
@@TMPChem Very true, I am in a similar situation and I am finding these videos incredibly helpful as even the literature contains limited descriptions about Quantum Chemistry.
Hi sir. Have some questions regarding Hartree Fock. Is it possible to solve a multi-electron atom-orbital with the Roothan equations? By assuming that an AO is a linear combination of a basic set, or does it apply only to MO? The book I am using in my course (Peter Atkins Molecular quantum mechanics) says that Roothaan is used for MO orbitals, while pure HF is used for atoms. My problem with this is that I believe it will be very difficult to change the AOs between each iteration, as you sort of mentioned in your video. But if one was to use pure HF, what would one use as initial atomic orbitals?
One final question about MO orbitals. When approximated as a linear combination of atomic orbitals, how does spin work? Does one multiply spin with the molecular orbital or is it fixed by itself when summarizing the linear combination of atomic orbitals (spin orbitals). I suppose you use spin molecule orbital in the Roothan equations, is this right?
Hi Sebastian. Let's address these questions one by one.
1). The word "orbital" inherently refers to one-electron wavefunctions. If we are solving for a wavefunction with multiple electrons we can use a Slater determinant of orbitals as an approximation to that multi-electron wavefunction, but it will be an approximation. That's what we're doing in HF-Roothaan, is minimizing the energy of a set of occupied orbitals.
2). A "molecular orbital" is an orbital in a molecule, whereas an "atomic orbital" is an orbital in an atom. Though when we say "atomic orbital" we often mean "atomic orbital basis function", which is used for the purposes of approximating a molecular orbital as a linear combination of these basis functions.
3). HF is used both for atoms and molecules. Any molecular system of more than one electron requires the use of approximate methods, and HF is one of the primary first approximations used to estimate the energy of both atomic and molecular systems.
4). As you mention, it is very difficult to solve the HF equations without some type of basis set. There are methods that can be used for very small systems, but which quickly become computationally intractible for sizeable systems. Essentially any method of numerically integrating a PDE can work in principal, though most common techniques are wildly inefficient.
5). Typically atomic HF uses AO basis functions, but only centered at a single nucleus, rather than at the multiple nuclei that would be in a molecule. The AO basis functions used for a carbon atom calculation typically wouldn't be any different than those used for a carbon atom within a molecular calculation.
6). Spin is an independent attribute of the spatial distribution of electrons, and is treated in the setup of the algorithm. My dedicated HF chapter in the Computational Chemistry playlist goes more in depth on the details of how spin is dealt with in the simplest HF flavor, restricted HF (RHF).
7). One can use spatial orbitals or spin orbitals in RHF, but spin is typically dealt with at the MO level, and is not attributed to AO basis functions.
So, there is a KxK F and S matrices for each individual atomic orbital, right?
A comment, it should be easier to follow if you'd omit the subscript '1'. The derivation is not only for electron 1
Hi Miguel. The K by K matrix is for *all* atomic orbitals and molecular orbitals. The rows are coefficients for a particular atomic orbital, and the columns are coefficients for a particular molecular orbital (for the C matrix anyway).
I have a question on "μ", does it refer to the basis functions of another AO? Thanks
F12 for example is an integral that depends on two different basis sets phi star 1 and phi 2. these are individual atomic orbital basis functions. he says that at 7:30
I'm not sure if he still answers questions to this day lol.
hi ,what i do exacly need is finding how to get the expression of each Fuv, Suv and coefficients c as well which u already menthioned them in your vedio u just have written their expression but how did they get to that i need their demonstration please sir
The actual values of the matrix elements are integrals whose formulas are described here, but not analytically computed. Such details are beyond the scope of this course, but are described thoroughly in "Molecular Electronic Structure Theory" by Helgaker ("the purple book" as it is often called). Some (but definitely not all) of these aspects will be surveyed in future Computational Chemistry videos.
To compute an arbitrary Fock matrix element requires specifying a form of the atomic orbital basis functions (typically linear combinations of Gaussian functions with various angular components). For standard Gaussian-type orbitals, these formulas are reasonably straightforward if you're only using s orbitals, but they get obscenely complicated as you climb the angular momentum ladder (p, d, f, g, h, ...). At very high angular momenta they are often approximated rather than using the recursion equations. You could find these formulas in Helgaker, but the naive implementation scales very poorly, so I would only recommend using them for low angular momenta (s, p, d) and/or for small toy applications.
the purple eqn on the right side of the note, don't we need another sum from μ=1 to K to represent K*K matrix?
The purple equation is equivalent to the following yellow equation. It's the yellow matrix multiplication equation for an individual row / column. The purple equation is true for all integer values of mu such that 1
then I guess I could understand it in the way I've written on my previous comment? (why'd you say "equivalent"?)
or did I get it wrong?
Whether I say A*B = C, or C_ij = sum(A_ik * B_kj), both are equivalent statements of the same matrix equation. The second equation is true for all 1
Thank you.
it's still uncomfortable to accept.
for me, the former one is a whole matrix and the latter one just a one of elements.
may i think that they're conventionally treated as same?
Maybe it should be specified in the video that k different purple equations are generated each with a different value of mu (- [1,k].
P.S.:Very good videos by the way
the problem is that our quantum teacher asked us to look for the proof of their expressions but i couldnt find any of them thx 4 yr help though
Sure thing. At this level, resources start becoming very scarce, and web searches are often not helpful. Many of the best electronic resources are often some random professor's online notes, as many of these things aren't formally published in the literature, and you have to know somebody who knows what book to look into. I had a similar situation 2 years ago, and it took me 9 months until I was finally able to ask somebody who knew what book to look in, and once I knew I looked it up I had the solution coded and passing unit tests in an hour. Sometimes 30 seconds of good advice is worth weeks of solo struggle.
@@TMPChem Very true, I am in a similar situation and I am finding these videos incredibly helpful as even the literature contains limited descriptions about Quantum Chemistry.