Chemistry - Why are basis sets needed?
Solution 1:
Spatial orbitals $\phi_i$ in modern electronic structure calculations are indeed typically expressed as a linear combination of a finite number of basis functions $\chi_k$, \begin{equation} \phi_i(1) = \sum\limits_{k=1}^{m} c_{ki} \chi_k(1) \, . \end{equation} In the early days, atomic orbitals were built out of basis functions, while molecular orbitals were built out of atomic orbitals, which is where the name of the approach, linear combination of atomic orbitals (LCAO) originated. Today both atomic and molecular orbitals are built out of basis functions, and while basis functions for molecular calculations are still typically centered on atoms, they are usually differ from the exact atomic orbitals due to approximations and simplifications. Besides, basis functions centered on bonds or lone pairs of electrons, or even plane waves, are also used as basis functions. Nevertheless, the approach is still commonly referred to as linear combination of atomic orbitals.
Now, to the very question of why do we do things this way?
On the one hand, the LCAO technique made its way into quantum chemistry as just another example of a widely used approach of reducing a complicated mathematical problem to the well-researched domain of linear algebra. To my knowledge, this was proposed first by Roothaan.1 However, and as it was mentioned by Roothaan from the beginning, the LCAO approach in present day quantum chemistry is also attractive from the general chemistry point of view: it is tempting to construct molecular orbitals in modern electronic structure theory from their atomic counterparts as it was done by Hund, Mulliken and others already in the early days of quantum theory2 and as it is though in high-school general chemistry courses today.
1) Roothaan, C.C.J. "New developments in molecular orbital theory." Reviews of modern physics 23.2 (1951): 69. DOI: 10.1103/RevModPhys.23.69
2) Pauling and Wilson in Introduction to Quantum Mechanics with Applications to Chemistry refer to the following works in that respect (p. 346):
F. Hund, Z. f. Phys. 51, 759 (1928); 73, 1 (1931); etc.; R. S. Mulliken, Phys. Rev. 32, 186, 761 (1928); 41, 49 (1932); etc.; M. Dunkel, Z. f. phys. Chem. B7, 81; 10, 434 (1930); E. Hückel ,Z.f. Phys. 60, 423 (1930); etc.
Hund's papers are unfortunately in German, but Mulliken's ones are quite an interesting read. Especially the second one,
Mulliken, Robert S. "Electronic structures of polyatomic molecules and valence. II. General considerations." Physical Review 41.1 (1932): 49. DOI: 10.1103/PhysRev.41.49
which (again, to my knowledge) introduced the very term "molecular orbital".
Solution 2:
The idea of using basis (in general sense, not just basis set) in electronic structure theory is due to the fact that we can't solve the wave function of a system analytically except those with only one electron (H, H2+,..). So, we need to introcude some approximations to find the wave function.
There are actually two levels of approximation:
1. Many-electron (or N-electron) approximation:
The wave function of N electrons (or a state of N electrons) can be approximately a Slater determinant (SD) or a linear combination of SD. A SD is an antisymmetric product of one-electron basis functions. These one-electron basis are called molecular orbitals or MO or orbital. An orbital is nothing but a state of one electron.
In sum, a many-electron wavefunction is constructed using orbitals which are one-electron states. That's what the name implies: many-electron (or N-electron) approximation
2. One-electron approximation:
- Now, the question is: we know that we need the orbitals to construct the wave function but how do we find those MOs or orbitals? Here, we introduce another approximation. We use a set of functions (called basis set) to construct the MOs. MOs are now constructed by a linear combination of those functions (or LCAO which is already explained very well by the above answer). The coefficients used in that construction are called MO coefficients, which are varied to minimize the energy of the corresponding many-electron wavefunction (the system of interest). Since one-electron state (MO) is constructed by some functions (basis set), this is called one-electron approximation. I don't like the term "atomic orbital" since they are just some mathematical functions. They are not even a wave function of one single atom. They can be "any function" providing it is convenient to work with. For example Gaussian function is often used since it is easier to calculate the integrals in comparison with Slater function. Plane wave is often used in periodic systems since the periodicity is conveniently imposed.
- In fact, we can use an infinite number of those functions to construct MOs (or orbitals), in that case, the calculation reaches the complete basis set (CBS) limit. In practice, we use a finite number of basis set, and many people came up with many types of basis set with different type of functions used (Gaussian, Slater, Muffin, numerical, or plane wave..) and with different sizes (minimum, double zeta, triple zeta,...)
In summary: In ab initio calculations, one has to provide these information:
Geometry of the systems (atom coordinates)
The total charge of the system
Spin state (singlet, doublet, triplet, ...)
A many-electron approximation (one SD, e.g. HF, DFT, MP2, CCSD OR many SDs like multiconfigurational methods, e.g. CI, MCSCF). This is also preferred as the level of correlation since different ways of constructing the wave function will recover different amounts of correlation (coulomb interaction between electrons). Methods using one SD like HF cannot capture the real interaction between electrons.
- A one-electron approximation (a basis set).
The entire information above is all we need to calculate electronic structure of a system, and this is known as a quantum chemical model defined by John Pople
Solution 3:
This answer, I hope, complements those earlier ones and give some little explanation as to how the basis set calculation works.
As pointed out already it is not necessary to use a basis set to solve the Schroedinger equation because for simple systems, such as a harmonic or Morse potential, analytical i.e. algebraic solutions are available. Numerical methods, such as the Numerov and Shooting methods, can be used for more complicated potentials but soon become intractable.
Generating a solution using a basis set is often computationally the best method, and sometimes the only method that can be used. This method involves solving the Schroedinger equation without direct integration.
The idea is to choose a known set of functions which when added together in the correct proportions will generate the wavefunction for the target problem. This is the same idea as using a fourier series to generate a target function; the more terms that are included in the series the better the fit to the target function becomes. An example of a fourier series is to form a polynomial using many sine/cosine waves of different amplitude and frequency.
In forming a basis set we choose a set of orthogonal functions, and numerically solve the matrix Schroedinger equation using as many of these functions as needed for the accuracy required. Normalised orthogonal functions have the property that $\int \phi^*_n\phi_mdx=\delta_{nm}$ where $\delta_{nm}$ is the Knonecker delta, if $n=m$ then $\delta_{nm}=1$ otherwise it is zero.
As an example we may use the sine-wave solutions to the particle in a box (PIB) problem to solve a harmonic well potential with a central 'hill' or barrier. If the PIB equation is written as $H\phi = E\phi$, the eigenvalues (energies) are $\epsilon_n=\int\phi^*_n~H~\phi_ndx$.
The new problem we want to solve has the equation
$$ H^{’} \psi = E^{’} \psi \tag {1} $$ which has a different Hamiltonian to reflect the different potential energy. The new wavefunction for the $n^{th}$ level is assumed to have the form
$$\psi_n=c_{1,n}\phi_1 + c_{2,n}\phi_2 \cdots c_{N,n}\phi_N$$
where the coefficients c are to be found. (The calculation follows that in J.Chem. Educ. 2012, v89, p1152). Perhaps twenty or a hundred terms in the basis set will be needed to obtain an accurate answer both for the eigenvalues (energies) and eigenvectors (wavefunctions). If $N \rightarrow \infty$ then the energy and wavefunctions would be exact.
The PIB basis set used here is a good starting point in our example as we know that the wavefunctions for a harmonic oscillator oscillate somewhat in a sinusoidal manner and the barrier harmonic well should be similar. If we used exponential functions many, many more would be needed to achieve the same result as these just do not have the same shape as the wavefunction. However, exponential functions would be good to use in describing an atoms wavefunctions as they have a shape similar to atomic orbital radial functions.
The energy for level m with the new Hamiltonian $H^{’}$ is found by left multiplying eqn. 1 by $\phi^*$
$$\int \phi^*_mH^{’} \psi_ndx =E^{’}_n\int \phi^*_m\psi_ndx$$ and substituting for $\psi$ gives $$\int \phi^*_mH^{’}(c_{1,n}\phi_1 + c_{2,n}\phi_2 \cdots)dx=E^{’}_n\int \phi^*_m(c_{1,n}\phi_1 + c_{2,n}\phi_2 \cdots)dx$$ On expanding this out and using the orthonormality the equation reduces to a matrix equation $$ \bf HC=E^{’}C$$ where $\bf H$ is a square matrix of elements $$H_{m,n} = \int \phi^*_mH^{’}\phi_ndx$$ and $\bf C$ is a column matrix of coefficients. The matrix equation is $\bf(H-E^{’}I)C=0$ where $\bf I$ is the unit diagonal matrix and $\bf 0$ a null matrix containing only zeros. Solving this equation involves evaluating many simple integrals $H_{m,n}$ in the secular determinant and then diagonalising to find the characteristic equation which is a polynomial in $E^{’}$ and hence the energies.