Paul,

main problem here is that chemists having quantum chemical courses usually lack basic knowledge from for example linear algebra and other mathematical disciplines. Explanation of mechanisms in organic chemistry and so on can operate with intuitively understandable concepts of effective charge, different kinds of affinities etc, but when one starts with quantum chemistry - everything becomes a bit complicated. Two functions may have equivalent modules, but be orthogonal. As they are spatially located in the same space, how to explain this without integrals?  And further amount of misunderstandings and faults grow rapidly: some researchers even do not understand the difference between MO and wave function, do not understand what is actually depicted in figures with MOs and so on. Then we have enormous amount of papers using Hartree Fock approximation as is as a main computational method, or applying B3LYP everywhere even though it's applicability is well known and numerous corrections or alternatives are available. 

Hi Moustafa!

If you recall the properties of linear differential equations, you may remember that multiplying an eigenfunction with any constant is again a solution to the equation. So, there is no fundamental mathematical problem in using wavefunctions that are not normalized - they still are eigenfunctions to your Hamiltonian. However, you would then have to introduce the normalization at a later stage when calculating some properties from that wavefunction, because -- as you suggest -- the probability of finding the electron anywhere in space must be 1 and your calculations need to obey that. So the easiest thing is to just include this in the wavefunction right away and not have to worry about it later.

Also, you were talking about "orthonormal" in the title of your question. Keep in mind that normalization and orthogonality are two different and independent properties of wavefunctions. (I fear orthogonality would be a bit more complicated to explain, unfortunately. But if you have questions, feel free to ask.)

As for orthogonality it is just a sequence of spectral theorem for hermitian operators in finite dimensional space. Hermitian operators always have a complete set of orthogonal eigenfunctions (at least up to a unitary transformation). As a solution of stationary schrodinger equation is an eigenfunction of hermitian operator, it will be orthogonal to all the other solutions.

Moustafa

If, as I suspect, you want an intuitive sense of orthogonality, this might help (forgive the rather elementary start, to ensure you can see easily what I’m trying to say):

σ and π- molecular orbital functions are independent, in the same sense as are atomic orbitals, i.e. no amount of an atomic s-orbital describes an atomic p-orbital, and further, no amount of e.g. a px-orbital describes a pz-  or py-orbital - these orbitals encode completely different-shaped/oriented electron-distributions in real space,

You can also think of the MO functions as each being encoded/represented by a vector in a high-dimensional "function space" (an MO function is a linear combination (LC) of the “basis functions”, and therefore can be viewed as a vector of coefficients forming the vector space). As the MOs are independent, then you can think of these vectors as being orthogonal (“at right angles to each other”) and (together with other conditions mentioned below) thus forming the vector space.

The problem can also be seen from the other direction: one starts to seek the MOs for a specific problem/molecule by choosing a suitable “basis set” of functions (think of them as representing atomic orbitals, AOs) that can form a vector space (the functions, e.g. gaussians that are independent of each other, can be multiplied by constants and added to each other, thus (again) constituting a vector space; the theoretician’s skill is in choosing a suitable basis set for the problem at hand).

In the variational method, the final (but still approximate) MO functions are also, as you know, eigensolutions of the hamiltonian operator, obtained by setting up the hamiltonian and constructing trial sets of MOs as linear combinations of the basis AOs, then solving the molecular problem via the minimisation of the energy expression ∫/ ∫ by iteratively modifying trial MOs ψ, until the iterative process converges to an acceptable solution.

You can probably now see how to view this process in vector-space terms, to see the role of orthogonality, as the energy minimisation results in a set of ψ functions, each of which is a linear combination of the original basis functions – i.e. the MOs are represented by vectors of the coefficients that multiply each of the component basis functions.

Now, here’s the final insight: you can think of the result of this variational process, of minimising the energy expression (once the iterative process has converged to an acceptable solution), as simply a rotation (possibly an improper rotation, that reverses some signs) of the original orthogonal basis vectors into a different orthogonal set of vectors that represent the solution MO functions in the vector space (the vectors’ lengths (“norms”) will usually also change prior to normalisation).

This rotation from the original orthogonal basis occurs as a result of the “diagonalization” of the energy matrix as the independence of the final  ψ vectors representing new solution MOs means that the off-diagonal elements of the matrix should all be zero and so each MO is a subspace of its own (i.e. a solution of the eigenvalue equation for that MO.)

So, the original basis functions are represented by orthogonal vectors, and the final MO solution functions are also represented by a (different) orthogonal set of vectors, and the two are related by a rotation of sorts.

As to the normality condition, as Matthias first explained above, this is just a matter of scaling the eigenfunction/eigenvector to ensure the correct probability.

Taking both orthogonality and normality conditions together, you should now be able to see that it’s most convenient to work with basis and solution sets that are ortho-normal.

As usual, Wikipedia offers more detail (please see link below).

Hope this helps – Paul

(For the purists, I must add that this is only a pedagogic attempt (for someone who might, like me, think in images), obviously not a formal layout of the theory. In spite of the risks, I suspect that many students find such an approach helpful at first. But if anyone sees a fatal flaw in this description, please don’t hesitate to correct me.)

http://en.wikipedia.org/wiki/Molecular_orbital

Thank you Matthias .

Thank you Oleg , but your answer is very advanced for me , I'm still beginner so i couldn't understand it ..

Thank you Paul , 

(As the MOs are independent, then you can think of these vectors as being orthogonal )

so,independent = in different planes ?

Moustafa,

I'll assume that when you ask "so,independent = in different planes ?" you are referring to the vectors in the vector space (not the orbitals in 3-space). As I visualise it, orthogonal vectors are perpendicular to the (hyper)planes containing all the other vectors.

Incidentally, I also couldn't make sense of  the phrase "orthogonality it is just a sequence of spectral theorem" that was given in one of the other replies.

Often, in academically-oriented social media, one comes across very smart people who can explain things from their own perspective without unpacking it for those who're not yet at their level (or, worst case, wish only to show how clever they are with comments that can come across to the rest of us as both unhelpful and dismissive).

While sites like ResearchGate and Academia.edu are intended mainly for discussion between active researchers, I think those who are willing to reply to a student should be satisfied to do so helpfully, for genuine seekers after understanding. Even replies on StackExchange, which is explicitly open to students as well, can sometimes be unhelpful. And Wikipedia should be oriented primarily to learners (as in "encyclopaedia") though some of the maths pages there still come across as having been written as concisely (~briefly) and unintelligibly as possible, by someone trying to show only their own virtuosity (skill); although I think Wikipedia has begun to improve on this lately. (This is a topic on which I have quite strong views.)

Paul

Here, independent does not mean in different planes but just like in three dimensional space i, j and k (called basis) are independent from each other and  any component of a vector can be found in in these three basis as Ax component or Ay component and Az component.... However, in Quantum mechanics instead of i, j and k we have finite or infinite basis represented by u1>, u2>,......... In simple words, all these basis behaves just like i, j and k... So any state vector which has all the information of the system can be solved for a particular basis which will give us the corresponding results we are looking for....... In my understanding each orbital can be represented by a basis and all the orbitals make a complete set........

Paul,

main problem here is that chemists having quantum chemical courses usually lack basic knowledge from for example linear algebra and other mathematical disciplines. Explanation of mechanisms in organic chemistry and so on can operate with intuitively understandable concepts of effective charge, different kinds of affinities etc, but when one starts with quantum chemistry - everything becomes a bit complicated. Two functions may have equivalent modules, but be orthogonal. As they are spatially located in the same space, how to explain this without integrals?  And further amount of misunderstandings and faults grow rapidly: some researchers even do not understand the difference between MO and wave function, do not understand what is actually depicted in figures with MOs and so on. Then we have enormous amount of papers using Hartree Fock approximation as is as a main computational method, or applying B3LYP everywhere even though it's applicability is well known and numerous corrections or alternatives are available. 

Thank you Dr.Iftikhar .

Dr.Oleg if you know some source for linear algebra or all aspects of mathematics with focus on QM examples , i will be thankful if you guided me to it . 

Moustafa, unfortunately I can't advise you anything as major part of literature I read as a student was in russian =)

Moustafa

While there’s only one book I’m aware of that’s likely to be precisely what you seek in your latest question (" some source for linear algebra or all aspects of mathematics with focus on QM examples", a few lines above this entry), I suggest you might first take a look at some of the many shorter sources, available free online, with varying degrees of emphasis on quantum physics:

(See first  six links below)

As I’m sure you know, there are also introductions to linear algebra in many QM textbooks. The texts for early-year students that seem to get the most frequent online recommendations are:

 -          R. Shankar’s Principles of Quantum Mechanics, (now in its 2nd Edition, Mar 16, 2011), which has a 57-page maths intro, mostly linear algebra, as the first chapter

-          David J. Griffiths’ Introduction to Quantum Mechanics (2nd edition) Hardcover – April 10, 2004 (just a few pages on LI in an appendix, but as it's a frequently-recommended book, you may find that a good start)

If you prefer a lecture/video format, tho’ it’s not oriented to QM, you might find the next link (Gilbert Strang) on linear algebra helpful.

FINALLY: When you feel ready to tackle the subject in depth, you might want to try the next link, Per-Olov Löwdin’s  “Linear Algebra for Quantum Theory”.

Although it's got a very high price for new copies, used copies are currently available through Amazon from £22.98.

Löwdin died in 2000, and I gather from AbeBooks (ninth link) that even Löwdin's  pre-print notes are now fetching good prices (clearly, I shall have to dig out my own copies when I have the time).

Curiously, I wasn’t aware of this book even though I did my post-doc under Löwdin at Kvantkemiska Institutionen in the 1970s.

Even more curiously, as you may see at the final link:

“Löwdin formulated in 1950 the symmetric orthogonalization scheme for molecular orbital calculations.”

So I guess he’s really the fellow to answer your original question in some depth.

Hope this helps – Paul

PS If you want worked examples of mathematics with focus on QM, there's also the "Schaum OuTline of Theory and Problems of Quantum Mechanics" by Peleg, Pnini and Zaarur.

(I’m gradually compiling a bibliography on quantum mechanics, for students and interested mature students, but it’s still some way – a year or few – away from publication.)

http://physics.stackexchange.com/questions/39165/linear-algebra-for-quantum-physics

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/751LinearAlgebra.pdf

https://www.math.uni-bielefeld.de/~hemion/Linear_Algebra_in_Physics/LinearAlg.Physics.pdf

http://www.physics.udel.edu/~msafrono/650/Lecture%204.pdf

http://www.astro.caltech.edu/~golwala/ph125ab/ph125_notes.pdf

http://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes2.pdf

http://ocw.mit.edu/faculty/gilbert-strang/

http://www.amazon.co.uk/Linear-Algebra-Quantum-Theory-Per-Olov/dp/0471199583

http://www.abebooks.co.uk/servlet/SearchResults?sts=t&tn=Linear+Algebra+for+Quantum+Theory

It is a honor to me to contact a professor like you Dr.Paul , thank you very much :)

Oleg,

I’m in fair agreement with you about theoretical chemistry being rather less intuitive than concepts relating to experimental chemistry, at least from the perspective of non-mathematicians.

And to support your point: I’m not familiar with modules (can’t find anything on them in relation to MOs, on the web - or was this point intended for Iftikhar?) so I’m guessing the AO / MO coefficients form algebraic rings, which suggests symmetry?

In regard to what I take as your 2nd point: Of course the potential for misunderstanding grows as one goes deeper into a subject. But then that’s partly my point too – the deeper it goes, the greater the need for clarity of explanation.

So let’s take your point about MOs and wavefunctions, which gave me pause for thought.

On the one hand: there’s the great difference between the way e.g. Huckel MOs are identified for pi-electron distributions, as compared with how the electron distribution is given in ab initio (full CI) calculations on a given chemical species. Of course, the latter approach does not use orbitals at all.

But on the other: they resemble each other, as both provide indications of the distribution of electric charge – which I see as the main characteristic of a wave-function, at least for theoretical chemistry.

So in what sense can one distinguish MOs and wave functions?

In non-relativistic cases, both implicitly involve a solution of the Schrodinger equation. Conceptually, perhaps, an MO description (as also a VB description) is a different “analysis”, or breakdown, of the electron distribution than might be obtained from a full CI solution – but with the exception of two-particle systems they’re anyway never analytic solutions, in either case.

My conclusion (and I’m open to further discussion on this) is that – except where one can have an analytic solution – in practical terms the difference between MOs and wavefunctions relates more to how one chooses to use the terminology than to any fundamental difference.

And where a difference in expertise exists between interlocutors, a more precise statement is of greater help to the lesser expert (and in this case, more than 40 years since being active in the field, that’s probably me!)

Thanks for providing an opportunity to review these matters – Paul

I think that a MO is a wave function (since it describes the electron distribution over the entire molecule) obtained by linear combination of the contributing wave functions of AOs , but the thing that I haven't understood yet is how it is concluded that these electrons represented by the LCAO are to interact or overlap , I read about the overlap matrix but i don't know how it is used exactly ?

Moustafa,

As Wikipedia says (1st link below):

"A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin. Necessarily this is an approximation, and highly accurate descriptions of the molecular electronic wave function do not have orbitals (see configuration interaction)." {NB: This also clarifies, a little more, the answer I gave previously.)

Putting it simply (and I hope correctly): the electrons interact, in a theoretical chemistry calculation, as such calculations are usually based on an approximation to the Schrodinger equation, which includes a potential energy term V (for e-e and e-nuc etc interactions) in the hamiltonian H. As you probably know already, finding the best LCAO representation of (approximate) MO wavefunctions via the energy-minimisation approach leads to terms of the type and , which get expanded in terms of the AOs, with the latter yielding the overlap matrix about which you've enquired.

As for the meaning of the overlap matrix, you might find the 2nd link, below, to be as good an explanation as you'll get.

I have to sign off from this whole conversation now. However, from the more recent questions you've been asking, it seems to me that you'll learn more from searching the web and textbooks yourself (almost all the answers are out there if you're prepared to work at it). Only raise questions here when you're really stuck.

Best of luck - Paul

http://en.wikipedia.org/wiki/Molecular_orbital

http://en.wikipedia.org/wiki/Orbital_overlap