Hückel method

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The Hückel method or Hückel molecular orbital method (HMO), proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of π electrons in conjugated hydrocarbon systems, such as ethylene, benzene and butadiene.^[1][2] It is the theoretical basis for Hückel's rule for the aromaticity of (4n + 2) π electron cyclic, planar systems. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon, known in this context as heteroatoms.^[3] A more dramatic extension of the method to include σ electrons, known as the extended Hückel method, was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to test the Woodward–Hoffmann rules.^[4]

In spite of its simplicity, the Hückel method in its original form makes qualitatively accurate and chemically useful predictions for many common molecules and is therefore a powerful educational tool. It is described in many introductory quantum chemistry and physical organic chemistry textbooks.

Hückel characteristics

The method has several characteristics:

• It limits itself to conjugated hydrocarbons.
• Only π electron molecular orbitals are included because these determine much of the chemical and spectral properties of these molecules. The σ electrons are assumed to form the framework of the molecule and σ connectivity is used to determine whether two π orbitals interact. However, the orbitals formed by σ electrons are ignored and assumed not to interact with π electrons. This is referred to as σ-π separability. It is justified by the orthogonality of σ and π orbitals in planar molecules. For this reason, the Hückel method is limited to systems that are planar or nearly so.
• The method is based on applying the variational method to linear combination of atomic orbitals and making simplifying assumptions regarding the overlap, resonance and Coulomb integrals of these atomic orbitals. It does not attempt to solve the Schrödinger equation, and neither the functional form of the basis atomic orbitals nor details of the Hamiltonian are involved.
• Interestingly, for hydrocarbons, the method takes atomic connectivity as the only input; empirical parameters are only needed when heteroatoms are introduced.
• The method predicts how many energy levels exist for a given molecule, which levels are degenerate and it expresses the molecular orbital energies in terms of two parameters, called α, the energy of an electron in a 2p orbital, and β, the interaction energy between two 2p orbitals (the extent to which an electron is stabilized by allowing it to delocalize between two orbitals). To understand and compare systems in a qualitative or even semi-quantitative sense, explicit numerical values for these parameters are typically not required.
• In addition the method also enables calculation of charge density for each atom in the π framework, the fractional bond order between any two atoms, and the overall molecular dipole moment.

Hückel results

The results for a few simple molecules are tabulated below:

Molecule	Energy	Frontier orbital	HOMO–LUMO energy gap
Ethylene	E1 = α – β	LUMO	−2β
	E2 = α + β	HOMO
Butadiene	E1 = α + 1.618...β	1.618... and 0.618... are ${\displaystyle ({\sqrt {5}}\pm 1)/2}$
	E2 = α + 0.618...β	HOMO	–1.236...β
	E3 = α – 0.618...β	LUMO
	E4 = α – 1.618...β
Benzene	E1 = α + 2β
	E2 = α + β
	E3 = α + β	HOMO	−2β
	E4 = α − β	LUMO
	E5 = α − β
	E6 = α − 2β
Cyclobutadiene	E1 = α + 2β
	E2 = α	SOMO	0
	E3 = α	SOMO
	E4 = α − 2β
Table 1. Hückel method results.  Lowest energies of top α and β are both negative values.[5] HOMO/LUMO/SOMO = Highest occupied/lowest unoccupied/singly-occupied molecular orbitals.

The theory predicts two energy levels for ethylene with its two π electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 π-electrons occupy 2 low energy molecular orbitals, out of a total of 4, and for benzene 6 energy levels are predicted, two of them degenerate.

For linear and cyclic systems (with N atoms), general solutions exist:^[6]

Frost circle (de) mnemonic for 1,3-cyclopenta-5-dienyl anion

• Linear system (polyene/polyenyl): ${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {(k+1)\pi }{N+1}}\quad (k=0,1,\ldots ,N-1)}$.
  • Energy levels are all distinct.

• Cyclic system, Hückel topology (annulene/annulenyl): ${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {2k\pi }{N}}\quad (k=0,1,\ldots ,\lfloor N/2\rfloor )}$.
  • Energy levels ${\displaystyle k=1,\ldots ,\lceil N/2\rceil -1}$ are each doubly degenerate.
• Cyclic system, Möbius topology (hypothetical^[7]): ${\displaystyle E_{k}=\alpha +2\beta '\cos {\frac {(2k+1)\pi }{N}},\ \beta '=\beta \cos(\pi /N)\quad (k=0,1,\ldots ,\lceil N/2\rceil -1)}$.
  • Energy levels ${\displaystyle k=0,\ldots ,\lfloor N/2\rfloor -1}$ are each doubly degenerate.

The energy levels for cyclic systems can be predicted using the Frost circle (de) mnemonic (named after the American chemist Arthur Atwater Frost (de)). A circle centered at α with radius 2β is inscribed with a polygon with one vertex pointing down; the vertices represent energy levels with the appropriate energies.^[8] A related mnemonic exists for linear and Möbius systems.^[9]

The value for β in Hückel theory is roughly constant for structurally similar compounds, but not surprisingly, structurally dissimilar compounds will give very different values for β. For example, using the π bond energy of ethylene (65 kcal/mole) and comparing the energy of a doubly-occupied π orbital (2α + 2β) with the energy of two isolated p orbitals (2α), a value of β = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene (36 kcal/mole, derived from heats of hydrogenation) and comparing benzene (6α + 8β) with a hypothetical "non-aromatic 1,3,5-cyclohexatriene" (6α + 6β), a much smaller value of β = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene (1.33 Å) compared to benzene (1.40 Å). The shorter distance between the interacting p orbitals accounts for the greater energy of interaction, which is reflected by a higher value of β. Nevertheless, Hückel theory should not be expected to provide accurate quantitative predictions; only semi-quantitative and qualitative trends and patterns are reliable.

With this caveat in mind, many predictions of the theory have been experimentally verified:

• The HOMO–LUMO gap, in terms of the β constant, correlates directly with the respective molecular electronic transitions observed with UV/VIS spectroscopy. For linear polyenes, the energy gap is given as:

${\displaystyle \Delta E=-4\beta \sin {\frac {\pi }{2(n+1)}}}$

from which a value for β can be obtained between −60 and −70 kcal/mol (−250 to −290 kJ/mol).^[10]

• The predicted molecular orbital energies as stipulated by Koopmans' theorem correlate with photoelectron spectroscopy.^[11]
• The Hückel delocalization energy correlates with the experimental heat of combustion. This energy is defined as the difference between the total predicted π energy (in benzene 8β) and a hypothetical π energy in which all ethylene units are assumed isolated, each contributing 2β (making benzene 3 × 2β = 6β).
• Molecules with molecular orbitals paired up such that only the sign differs (for example α ± β) are called alternant hydrocarbons and have in common small molecular dipole moments. This is in contrast to non-alternant hydrocarbons, such as azulene and fulvene that have large dipole moments. The Hückel theory is more accurate for alternant hydrocarbons.
• For cyclobutadiene the theory predicts that the two high-energy electrons occupy a degenerate pair of molecular orbitals that are neither stabilized or destabilized. Hence the square molecule would be a very reactive triplet diradical (the ground state is actually rectangular without degenerate orbitals). In fact, all cyclic conjugated hydrocarbons with a total of 4nπ electrons share this molecular orbital pattern, and this forms the basis of Hückel's rule.
• Dewar reactivity numbers deriving from the Hückel approach correctly predict the reactivity of aromatic systems with nucleophiles and electrophiles.

Mathematics behind the Hückel method

The mathematics of the Hückel method is based on the Ritz method. In short, given a basis set of n normalized atomic orbitals ${\displaystyle \{\phi _{i}\}_{i=1}^{n}}$, an ansatz molecular orbital ${\displaystyle \psi _{g}=N(c_{1}\phi _{1}+\cdots +c_{n}\phi _{n})}$ is written down, with normalization constant N and coefficients ${\displaystyle c_{i}}$ which are to be determined. In other words, we are assuming that the molecular orbital (MO) can be written as a linear combination of atomic orbitals, a conceptually intuitive and convenient approximation (the linear combination of atomic orbitals or LCAO approximation). The variational theorem states that given an eigenvalue problem ${\displaystyle {\hat {H}}\psi ^{(i)}=E^{(i)}\psi ^{(i)}}$ with smallest eigenvalue ${\displaystyle E^{(0)}}$ and corresponding wavefunction ${\displaystyle \psi ^{(0)}}$, any normalized trial wavefunction ${\displaystyle \psi _{g}}$ (i.e., ${\textstyle \int _{\mathbb {R} ^{3}}\psi _{g}^{*}\,\psi _{g}\,dV=1}$ holds) will satisfy

${\displaystyle {\mathcal {E}}[\psi _{g}]=\int _{\mathbb {R} ^{3}}\psi _{g}^{*}\,{\hat {H}}\psi _{g}\,dV\geq E^{(0)}}$,

with equality holding if and only if ${\displaystyle \psi _{g}=\psi ^{(0)}}$. Thus, by minimizing ${\displaystyle E(c_{1},\ldots ,c_{n})={\mathcal {E}}[\psi _{g}]}$ with respect to coefficients ${\displaystyle c_{i}}$ for normalized trial wavefunctions ${\displaystyle \psi _{g}(c_{1},\ldots ,c_{n})}$, we obtain a closer approximation of the true ground-state wavefunction and its energy.

To start, we apply the normalization condition to the ansatz and expand to get an expression for N in terms of the ${\displaystyle c_{i}}$. Then, we substitute the ansatz into the expression for E and expand, yielding

${\displaystyle E(c_{1},\ldots ,c_{n})=N^{2}{\Big [}\sum _{i=1}^{n}c_{i}^{2}H_{ii}+\sum _{1\leq i\neq j\leq n}c_{i}c_{j}H_{ij}{\Big ]}}$, where ${\displaystyle N={\Big [}\sum _{i=1}^{n}c_{i}^{2}S_{ii}+\sum _{1\leq i\neq j\leq n}c_{i}c_{j}S_{ij}{\Big ]}^{-1/2}}$,

${\displaystyle S_{ij}=\int _{\mathbb {R} ^{3}}\phi _{i}^{*}\,\phi _{j}\,dV}$, and ${\displaystyle H_{ij}=\int _{\mathbb {R} ^{3}}\phi _{i}^{*}\,{\hat {H}}\phi _{j}\,dV}$.

In the remainder of the derivation, we will assume that the atomic orbitals are real. (For the simple case of the Hückel theory, they will be the 2p_z orbitals on carbon.) Thus, ${\displaystyle S_{ij}=S_{ji}^{*}=S_{ji}}$, and because the Hamiltonian operator is hermitian, ${\displaystyle H_{ij}=H_{ji}^{*}=H_{ji}}$. Setting ${\displaystyle \partial {E}/\partial {c_{i}}=0}$ for ${\displaystyle i=1,\ldots ,n}$ to minimize E and collecting terms, we obtain a system of n simultaneous equations

${\displaystyle \sum _{j=1}^{n}c_{j}(H_{ij}-ES_{ij})=0\quad (i=1,\cdots ,n)}$.

When ${\displaystyle i\neq j}$, ${\displaystyle S_{ij}}$ and ${\displaystyle H_{ij}}$ are called the overlap and resonance (or exchange) integrals, respectively, while ${\displaystyle H_{ii}}$ is called the Coulomb integral, and ${\displaystyle S_{ii}=1}$ simply expresses that fact that the ${\displaystyle \phi _{i}}$ are normalized. The n × n matrices ${\displaystyle [S_{ij}]}$ and ${\displaystyle [H_{ij}]}$ are known as the overlap and Hamiltonian matrices, respectively.

By a well-known result from linear algebra, finding nontrivial solutions to the simultaneous equations can be achieved by finding values of ${\displaystyle E}$ such that

${\displaystyle \mathrm {det} ([H_{ij}-ES_{ij}])=0}$. (*)

This determinant expression is known as the secular determinant. The variational theorem guarantees that the lowest value of ${\displaystyle E}$ that gives rise to a nontrivial (that is, not all zero) solution set ${\displaystyle (c_{1},c_{2},\ldots ,c_{n})}$ represents the best LCAO approximation of the energy of the most stable π orbital; higher values of ${\displaystyle E}$ with nontrivial solution sets represent reasonable estimates of the energies of the remaining π orbitals.

The Hückel method makes a few further simplifying assumptions concerning the values of the ${\displaystyle S_{ij}}$ and ${\displaystyle H_{ij}}$. In particular, it is first assumed that distinct ${\displaystyle \phi _{i}}$ have zero overlap. Together with the assumption that ${\displaystyle \phi _{i}}$ are normalized, this means that the overlap matrix is the n × n identity matrix: ${\displaystyle [S_{ij}]=\mathbf {I} _{n}}$. The generalized eigenvalue problem (*) then reduces to finding the eigenvalues of the Hamiltonian matrix ${\displaystyle [H_{ij}]}$.

Second, in the simplest case of a planar, unsaturated hydrocarbon, the Hamiltonian matrix ${\displaystyle \mathbf {H} =[H_{ij}]}$ is parameterized in the following way:

${\displaystyle H_{ij}={\begin{cases}\alpha ,&i=j;\\\beta ,&i,j\ \ {\text{adjacent}};\\0,&{\text{otherwise}}.\end{cases}}}$ (**)

Thus, we are assuming that: (1) the energy of an electron in an isolated C(2p_z) orbital is ${\displaystyle H_{ii}=\alpha }$; (2) the energy of interaction between C(2p_z) orbitals on adjacent carbons i and j (i.e., i and j are connected by a σ-bond) is ${\displaystyle H_{ij}=\beta }$; (3) orbitals on carbons not joined in this way are assumed not to interact, so ${\displaystyle H_{ij}=0}$ for nonadjacent i and j; and, as mentioned above, (4) the spatial overlap of electron density between different orbitals, represented by non-diagonal elements of the overlap matrix, is ignored by setting ${\displaystyle S_{ij}=0\ \ (i\neq j)}$, even when the orbitals are adjacent.

This neglect of orbital overlap is an especially severe approximation. For typical bond distances (1.40 Å) as might be found in benzene, for example, the true value of the overlap for C(2p_z) orbitals on adjacent atoms i and j is about ${\displaystyle S_{ij}=0.21}$; even larger values are found when the bond distance is shorter (e.g., ${\displaystyle S_{ij}=0.27}$ ethylene).^[12] A major consequence of having nonzero overlap integrals is the fact that, compared to non-interacting isolated orbitals, bonding orbitals are not energetically stabilized by nearly as much as antibonding orbitals are destabilized. The orbital energies derived from the Hückel treatment do not account for this asymmetry (see Hückel solution for ethylene (below) for details).

The eigenvalues of ${\displaystyle \mathbf {H} }$ are the Hückel molecular orbital energies ${\displaystyle E_{1},\ldots ,E_{n}}$, expressed in terms of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, while the eigenvectors are the Hückel MOs ${\displaystyle \Psi _{1},\ldots ,\Psi _{n}}$, expressed as linear combinations of the atomic orbitals ${\displaystyle \phi _{i}}$. Using the expression for the normalization constant N and the fact that ${\displaystyle [S_{ij}]=\mathbf {I} _{n}}$, we can find the normalized MOs by incorporating the additional condition

${\displaystyle \sum _{i=1}^{n}c_{i}^{2}=1}$.

The Hückel MOs are thus uniquely determined when eigenvalues are all distinct. When an eigenvalue is degenerate (two or more of the ${\displaystyle E_{i}}$ are equal), the eigenspace corresponding to the degenerate energy level has dimensionality greater than 1, and the normalized MOs at that energy level are then not uniquely determined. When that happens, further assumptions pertaining to the coefficients of the degenerate orbitals (usually ones that make the MOs orthogonal and mathematically convenient^[13]) have to be made in order to generate a concrete set of molecular orbital functions.

If the substance is a planar, unsaturated hydrocarbon, the coefficients of the MOs can be found without appeal to empirical parameters, while orbital energies are given in terms of only ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. On the other hand, for systems containing heteroatoms, such as pyridine or formaldehyde, values of correction constants ${\displaystyle h_{\mathrm {X} }}$ and ${\displaystyle k_{\mathrm {X-Y} }}$ have to be specified for the atoms and bonds in question, and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ in (**) are replaced by ${\displaystyle \alpha +h_{\mathrm {X} }\beta }$ and ${\displaystyle k_{\mathrm {X-Y} }\beta }$, respectively.

Hückel solution for ethylene

Molecular orbitals ethylene ${\displaystyle E=\alpha -\beta }$

In the Hückel treatment for ethylene, we write the Hückel MOs ${\displaystyle \Psi \,}$ as a linear combination of the atomic orbitals (2p orbitals) on each of the carbon atoms:

${\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}}$.

Molecular orbitals ethylene ${\displaystyle E=\alpha +\beta }$

Applying the result obtained by the Ritz method, we have the system of equations

${\displaystyle {\begin{bmatrix}H_{11}-ES_{11}&H_{12}-ES_{12}\\H_{21}-ES_{21}&H_{22}-ES_{22}\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$,

where:

${\displaystyle H_{ij}=\int \phi _{i}\,{\hat {H}}\phi _{j}\,dV}$ and

${\displaystyle S_{ij}=\int \phi _{i}\,\phi _{j}\,dV}$.

(Since 2p_z atomic orbital can be expressed as a pure real function, the * representing complex conjugation can be dropped.) The Hückel method assumes that all overlap integrals (including the normalization integrals) equal the Kronecker delta, ${\displaystyle S_{ij}=\delta _{ij}\,}$, all Coulomb integrals ${\displaystyle H_{ii}\,}$ are equal, and the resonance integral ${\displaystyle H_{ij}\,}$ is nonzero when the atoms i and j are bonded. Using the standard Hückel variable names, we set

${\displaystyle H_{11}=H_{22}=\alpha \,}$,

${\displaystyle H_{12}=H_{21}=\beta \,}$,

${\displaystyle S_{11}=S_{22}=1\,}$, and

${\displaystyle S_{12}=S_{21}=0\,}$.

The Hamiltonian matrix is

${\displaystyle \mathbf {H} ={\begin{bmatrix}\alpha &\beta \\\beta &\alpha \\\end{bmatrix}}}$.

The matrix equation that needs to be solved is then

${\displaystyle {\begin{bmatrix}\alpha -E&\beta \\\beta &\alpha -E\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$,

or, dividing by ${\displaystyle \beta }$,

${\displaystyle {\begin{bmatrix}{\frac {\alpha -E}{\beta }}&1\\1&{\frac {\alpha -E}{\beta }}\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$.

Setting ${\displaystyle x:={\frac {\alpha -E}{\beta }}}$, we obtain

${\displaystyle {\begin{bmatrix}x&1\\1&x\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$. (***)

This homogeneous system of equations have nontrivial solutions for ${\displaystyle c_{1},c_{2}}$ (solutions besides the physically meaningless ${\displaystyle c_{1}=c_{2}=0}$) iff the matrix is singular and the determinant is zero:

${\displaystyle {\begin{vmatrix}x&1\\1&x\\\end{vmatrix}}=0}$.

Solving for ${\displaystyle x}$,

${\displaystyle x^{2}-1=0\,}$, or

${\displaystyle x=\pm 1\,}$.

Since ${\displaystyle E=\alpha -x\beta }$, the energy levels are

${\displaystyle E=\alpha -\pm 1\times \beta }$, or

${\displaystyle E=\alpha \mp \beta }$.

The coefficients can then be found by expanding (***):

${\displaystyle c_{2}=-xc_{1}\,}$and

${\displaystyle c_{1}=-xc_{2}\,}$.

Since the matrix is singular, the two equations are linearly dependent, and the solution set is not uniquely determined until we apply the normalization condition. We can only solve for ${\displaystyle c_{2}}$ in terms of ${\displaystyle c_{1}}$:

${\displaystyle c_{2}=-\pm 1\times c_{1}\,}$, or

${\displaystyle c_{2}=\mp c_{1}\,}$.

After normalization with ${\displaystyle c_{1}^{2}+c_{2}^{2}=1}$, the numerical values of ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ can be found:

${\displaystyle c_{1}={\frac {1}{\sqrt {2}}}}$ and ${\displaystyle c_{2}=\mp {\frac {1}{\sqrt {2}}}}$.

Finally, the Hückel molecular orbitals are

${\displaystyle \Psi _{\mp }=c_{1}\phi _{1}+c_{2}\phi _{2}={\frac {1}{\sqrt {2}}}\phi _{1}\mp {\frac {1}{\sqrt {2}}}\phi _{2}={\frac {\phi _{1}\mp \phi _{2}}{\sqrt {2}}}\,}$.

The constant β in the energy term is negative; therefore, ${\displaystyle E_{+}=\alpha +\beta }$ with ${\textstyle \Psi _{+}={\frac {1}{\sqrt {2}}}(\phi _{1}+\phi _{2})\,}$ is the lower energy corresponding to the HOMO energy and ${\displaystyle E_{-}=\alpha -\beta }$ with ${\textstyle \Psi _{-}={\frac {1}{\sqrt {2}}}(\phi _{1}-\phi _{2})\,}$ is the LUMO energy.

If, contrary to the Hückel treatment, a positive value for ${\displaystyle S:=S_{12}=S_{21}}$ were included, the energies would instead be

${\displaystyle E_{\pm }={\frac {\alpha \pm \beta }{1\pm S}}}$,

while the corresponding orbitals would take the form

${\displaystyle \Psi _{\pm }={\sqrt {\frac {1}{2\pm 2S}}}\phi _{1}\pm {\sqrt {\frac {1}{2\pm 2S}}}\phi _{2}}$.

An important consequence of setting ${\displaystyle S>0}$ is that the bonding (in-phase) combination is always stabilized to a lesser extent than the antibonding (out-of-phase) combination is destabilized, relative to the energy of the free 2p orbital. Thus, in general, 2-center 4-electron interactions, where both the bonding and antibonding orbitals are occupied, are destabilizing overall. This asymmetry is ignored by Hückel theory. In general, for the orbital energies derived from Hückel theory, the sum of stabilization energies for the bonding orbitals is equal to the sum of destabilization energies for the antibonding orbitals, as in the simplest case of ethylene shown here and the case of butadiene shown below.

Hückel solution for 1,3-butadiene

Butadiene molecular orbitals

Similarly, in the Hückel treatment for 1,3-butadiene, we write the molecular orbital ${\displaystyle \Psi \,}$ as a linear combination of the four atomic orbitals ${\displaystyle \phi _{i}}$ (carbon 2p orbitals) with coefficients ${\displaystyle c_{i}}$:

${\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}+c_{3}\phi _{3}+c_{4}\phi _{4}}$.

The Hamiltonian matrix is

${\displaystyle \mathbf {H} ={\begin{bmatrix}\alpha &\beta &0&0\\\beta &\alpha &\beta &0\\0&\beta &\alpha &\beta \\0&0&\beta &\alpha \\\end{bmatrix}}}$.

In the same way, we write the secular equations in matrix form as

${\displaystyle {\begin{bmatrix}\alpha -E&\beta &0&0\\\beta &\alpha -E&\beta &0\\0&\beta &\alpha -E&\beta \\0&0&\beta &\alpha -E\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\\c_{4}\\\end{bmatrix}}=0}$,

which leads to

${\displaystyle {\Big (}{\frac {\alpha -E}{\beta }}{\Big )}^{4}-3{\Big (}{\frac {\alpha -E}{\beta }}{\Big )}^{2}+1=0}$

and

${\displaystyle E_{1,2,3,4}=\alpha +{\frac {{\sqrt {5}}\pm 1}{2}}\beta ,\alpha -{\frac {{\sqrt {5}}\mp 1}{2}}\beta }$, or approximately,

${\displaystyle E_{1,2,3,4}\approx \alpha +1.618\beta ,\alpha +0.618\beta ,\alpha -0.618\beta ,\alpha -1.618\beta }$, where 1.618... and 0.618... are the golden ratios ${\displaystyle \varphi }$ and ${\displaystyle 1/\varphi }$.

The orbitals are given by

${\displaystyle \Psi _{1}\approx 0.372\phi _{1}+0.602\phi _{2}+0.602\phi _{3}+0.372\phi _{4}}$,

${\displaystyle \Psi _{2}\approx 0.602\phi _{1}+0.372\phi _{2}-0.372\phi _{3}-0.602\phi _{4}}$,

${\displaystyle \Psi _{3}\approx 0.602\phi _{1}-0.372\phi _{2}-0.372\phi _{3}+0.602\phi _{4}}$, and

${\displaystyle \Psi _{4}\approx 0.372\phi _{1}-0.602\phi _{2}+0.602\phi _{3}-0.372\phi _{4}}$.