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Correct form of the electron wavefunction in periodic solids

The Bloch wavefunction leads either to mathematically impossible consequences or suggests that the ground state energy is a function of size and shape when the geometry of large crystals is considered in detail. It is incompatible with the assumption underlying the Born-von Karman periodic boundary condition. The source of the difficulty is the incorrect dependence of the Bloch wavefunction on the wavenumber index k. The mathematically impossible consequences can be overcome if the periodic part of the electron wavefunction is represented as un(r), which is dependent only on the band index, n, and is independent of the wavenumber index k. This correct form of the wavefunction is consistent with the Bloch theorem and with all other properties of Bloch wavefunctions. The correct form is also consistent with the Born-von Karman periodic boundary condition. The correct form of the electronic wavefunction in a periodic solid has profound consequences. It simplifies the calculation of electronic structure as only one wavefunction per band, un(r), needs to be evaluated. It brings about a conceptual unification between the band picture favored by physicists and the bond picture favored by chemists. The correct form of the electron wavefunction will simplify the understanding of many phenomena involving valence electrons.

The Bloch wavefunction is the starting point for the application of Quantum Mechanics to periodic solids. The Bloch Theorem imposes conditions on the form of the electron wavefunction in periodic solids. This theorem states that due to the translational periodicity of the Bravais lattice (and the potential), the wavefunction of the electron has the same periodicity up to a phase factor and can be represented as theorem is proved in different ways in standard textbooks [1][2][3]. Various other properties of the Bloch wavefunction are also described in standard textbooks [1,2].
In this paper, we show that the Bloch wavefunction, Eq.2, leads to mathematically impossible consequences or suggests that the ground state energy is a function of size and shape when the geometry of crystals is considered in detail. It is incompatible with the assumption underlying the Born-von Karman periodic boundary condition. The source of the difficulty is the incorrect dependence of the Bloch wavefunction on the wavenumber The Born-von Karman periodic boundary condition [1,3] is given by The where b i are the reciprocal lattice vectors [3]. Also, / where N i are the total number of unit cells in each dimension and n i take integer values from 0 to N i -1. This leads to the well known [1,3] result that the number of k points in a Brilloiun Zone equals the total number of unit cells, N, in the crystal. The set of wavenumbers {k} is uniquely determined by the size and shape [1,3] of a crystal. The electron is delocalized over the entire crystal. Therefore, the normalization condition of Bloch wavefunction, Eq.2, yields bulk properties of large crystals are unaffected by their size and shape, the electron density for band n is given by the sum Eq.5 represents the expression for charge density in crystals with different size, shape and number of unit cells, N, N′, N″ etc. In Eq.5, we have implicitly assumed that the band is fully occupied, but the arguments are independent of this assumption and are equally valid for partially filled bands. Each summation contains only positive terms. Eq.5 contains an infinite number of equalities, one for each size and shape of a crystal. The Bloch wavefunction, Eq.2, implicitly assumes that the periodic part of the wavefunction, ( ) n u k r , is different and distinct for different values of wavenumber, k. Otherwise, there would be no need for the wavenumber index, k, to be part of the description of ( ) n u k r .
This dependence of ( ) n u k r on the wavenumber index, k, leads to mathematically impossible consequences as discussed below.
We consider (without loss of generality) cubic crystals whose dimensions are more than 1-2 μm in each dimension to avoid size effects prevalent in nanometer size crystals. An average lattice parameter is of the order of 3-4 A, and hence, we consider a minimum number of 4000 or 2 12 unit cells along each dimension in a crystal. Therefore, a three dimensional crystal is considered to have a minimum number of 64 10 9 or 2 36 unit cells.
However, there is no theoretical upper limit on the number of unit cells in a crystal. Many single crystals, e.g. Si, are routinely grown to sizes of inches and feet. Here, a reasonable upper limit for the size of a crystal is considered to be higher by approximately four orders of magnitude in each dimension than the lower limit, which would make each dimension to be of the order of a few centimeters. Therefore, in such a large crystal 64 10 21 or 2 75 unit cells would be present. We consider the consequences of Eq.5 on different crystals below.
Case A -Two crystals of identical shape but different sizes. Consider 2 large cubic crystals (C 1 and C 2 ) with dimensions in the centimeter range or 2 25 unit cells in each dimension. If in one of these crystals, C 2 , each dimension is repeatedly halved and new crystals labeled C 2′ , C 2″ , etc. then the number of unit cells is reduced by ½ 3 for each halving. Since the number of unit cells in each dimension, N i , is reduced, the number of k points is also reduced appropriately. However, it is important to note that no new k points are generated, i.e. only 1 of every 8 k points is retained for each halving. According to Eq.5, the charge density, ( ) n r ρ , is exactly the same in both crystals (C 1 and C 2′ ) even though the summation in Eq.5 is over different number of terms. The summation for crystal C 1 contains all the same terms as crystal C 2′ but contains 7 times additional terms that are all different yet both summations give exactly the same charge density, adjusted for normalization. This process can be continued and the second crystal can be halved till each dimension is 2 12 unit cells (in the μm range) and this crystal can be labeled C 2′″ .
Comparison of crystals C 1 and C 2′″ show that exactly the same charge density must be obtained by the summation given by Eq.5. Again, the k points in C 2′″ are a subset of those in crystal C 1 and no new k points are introduced. According to Eq.5, the charge density for C 2′″ contains a summation over 2 36 different terms whereas for C 1 contains a summation over 2 75 different terms of which 2 36 are the same terms as in C 2′″ .
Considering that each summation contains only positive non-zero terms, it is impossible to conceive of any type of mathematical functions that can give exactly the same charge density (adjusted for normalization) when astronomical numbers of different functions are summed over their magnitude squared.
Case B -Two crystals of identical size but different shapes. Consider two crystals, C 3 and C 4 , with the same number of unit cells, N, but different shapes due to which the set of wavenumbers {k} will be different for each crystal. As in Eq.5, the charge density is expressed as Theorem [5] that the ground state energy of large crystals will depend on their size and shape. Therefore, it follows that all bulk properties of large crystals will depend on their size and shape, which is exactly opposite to the fundamental assumption underlying Born-von Karman periodic boundary conditions. Therefore, the Bloch wavefunction is incompatible with Born-von Karman periodic boundary condition. Clearly both of them cannot be correct and one of them is in error. The assumption underlying Born-von Karman periodic boundary condition, viz. bulk properties of large crystals are independent of size and shape, is unlikely to be in error as it is difficult to conclude that the charge density within a unit cell deep inside a large crystal will be affected by its size and shape. If it were true, it would suggest that ground state energy and bulk properties of large crystals must be specified with reference to their size and shape. This is not supported by any experimental observations of bulk properties. Hence, any error in the assumption underlying Born-von Karman periodic boundary condition can be ruled out.
Therefore, the likely source of error is in the Bloch wavefunction. We see from applying Eq.5 to cases A, B and C above that the Bloch wavefunction, Eq.2, leads to mathematically impossible consequences. Since That is, the periodic part of the wavefunction is identical for all values of wavenumber k.
By simple substitution it is readily seen that this wavefunction satisfies the Bloch Theorem, Eq.1. It also satisfies all other properties of electronic wavefunction in periodic solids, i.e. properties required of Bloch wavefunction, Eq.2, discussed in standard textbooks [1,2]. This is only to be expected as the correct electronic wavefunction, Eq.6, is a special case of the Bloch wavefunction, Eq.2, where all ( ) ( ) n n u u = k r r and will therefore satisfy all conditions that Bloch wavefunction satisfies. This form of the electron wavefunction readily overcomes the conditions (Cases -A, B and C) imposed by the geometry of crystals described above. This is because each term in every summation in Eq.5 is identical, i.e. Therefore, at least a plausible justification if not a rigorous mathematical proof for the existence of mathematical functions that would satisfy cases A, B and C discussed above is necessary without which the use of Bloch wavefunction is theoretically unjustified.
The correct form of the electronic wavefunction, Eq.6, has profound consequences. When Eq.6 is substituted for the electron wavefunction, the Schrodinger's equation for periodic solids is modified to where n ε k is the energy level of the electron wavefunction, ( ) n ψ k r , given by Eq.6. The first point to note is that it is only necessary to solve for one function per band, ( ) The potential energy is given by PE = E e-e + E n-e + E xc + E n-n where all the terms have their usual meaning. The nuclear-nuclear repulsion term, E n-n , is usually evaluated by the Ewald summation technique [3] that uses artificial parameters. Recently, we have described the correct method [6] to evaluate this term by incorporating zero point vibrations. The electron-electron repulsion energy term between two electrons is given by This shows that the electron-electron repulsion energy between two electrons is independent of k. It is readily seen that the nuclear-electron repulsion energy is also independent of k. Therefore, it is necessary to calculate E e-e and E n-e only once for each combination of bands, n and n', unlike in the Bloch form of the wavefunction, Eq.2, where they have to be calculated for each value of k separately. This also leads to an enormous reduction in computational effort.
The exchange energy between all electrons in bands n and n', when the Bloch form, Eq.2, is used is given by Eq.12 shows that the expression for exchange energy in periodic solids between all electrons in bands n and n' is very similar to that in atoms and molecules (where n and n' would represent energy levels) but differs in that the integral contains an additional function f(r i ,r j ). This additional function is due to the presence of many electrons in a given band. Therefore, evaluating the exchange energy in the local density approximation (LDA) using the same approximation ( 1/ 3 ( ) ex E ρ ∝ r ) for both molecules and solids [7,8] is not appropriate. Another interesting possibility is that since for some molecules with small number of electrons it is possible to evaluate the exchange energy integral, Eq.12 suggests that the same may be possible for some solids as well.
In while only the phase factor e ik.r will be altered. Therefore, to first order, the coulomb interaction energy terms remain unchanged as seen from Eq.9. The change in electron energy on excitation is dominated by the kinetic energy terms with a (small) contribution from the exchange energy term. In other words, the additional energy acquired by the electron upon excitation is mostly in the form of kinetic energy. Clearly, there will be many phenomena involving valence electrons whose explanations will be simplified when the correct form of the wavefunction, Eq.6, is used.
In conclusion, the Bloch wavefunction leads either to mathematically impossible consequences or suggests that the ground state energy is a function of size and shape