Molecular orbital method

Abbreviated as MO method. A type of approximation method for determining the electronic state of molecules. In contrast to this, there are approximation methods known as the atomic orbital method (AO method) or valence bond method (VB method). Even if Schrödinger's wave equation is applied to molecules with many electrons, it cannot be solved due to the addition of many-body and multi-center problems. For this reason, various approximation methods have been used, but the starting point of these ideas can be broadly divided into the MO method and the AO method. In both cases, the molecular orbital functions are usually constructed using atomic orbitals (atomic wave functions). In the AO method, molecules are considered from the standpoint that electrons are strongly bound to the constituent atoms, while in the MO method, electrons are considered to be free from atoms and spread throughout the molecule. In this way, the AO method emphasizes the individuality of the constituent atoms, but the MO method ignores this.
However, regardless of which method you start with, you can get a good agreement by improving the approximation. In the MO method, the orbitals that are spread throughout the molecule are assumed to resemble the atomic orbitals of the atoms close to the atoms, and the molecular orbitals are often approximated as a linear combination of the atomic orbitals (AOs) of the constituent atoms. This is called the LCAO approximation. The number of molecular orbitals (MOs) is equal to the number of AOs used, but each MO can accommodate up to two electrons with antiparallel spins due to the Pauli principle. The ground state is given by placing two electrons into each MO in order of decreasing energy. For example, in a hydrogen molecule, the ground state is created by placing two electrons with antiparallel spins into the bonding molecular orbital. In this case, since the electrons are moving throughout the molecule, there is a chance that two electrons will gather at one atom at the same time, resulting in an overestimation of the contribution of the ionic structure. To correct this, a method is used that introduces other states (excited states). This is called the configuration interaction (CI) approximation.
Among the LCAO approximation MO methods, the Hückel approximation is used as the simplest method that does not explicitly take CI into consideration. The features of this method are that, first, the Hamiltonian of a multi-electron system is expressed as the sum of effective single-electron Hamiltonians, with the inter-electron interactions being assigned equally to each electron. Next, to estimate the matrix elements obtained in this way, in the case of the π-electron systems of hydrocarbons, the Coulomb integral and resonance integral are regarded as empirical parameters, and are determined so as to reproduce the measured values. Although the Hückel method is not suitable for quantitative research, it is effective when comparing various molecules.
The Hückel MO method is only applicable to π-electron systems, but there is an extended Hückel method that handles σ-electron systems. In this method, LCAO-MOs are created using all atomic orbitals except for the core electrons (1s electrons), the Coulomb integral is estimated using the ionization potential of the valence state, and the resonance integral is calculated from the Coulomb integral and the overlap integral. The MO method that explicitly considers the electron-electron interactions is called the antisymmetric molecular orbital (ASMO) method. In the ASMO method, the wave function of the entire system, including the spin function, is antisymmetric so that it satisfies the Pauli principle. In general, except for special molecules with high symmetry, it is not possible to know the AO coefficients in the LCAO-MO from the beginning. In such cases, the MOs are determined using a procedure similar to that of the Hartree-Fock self-consistent field (SCF) method. The theoretical calculation results using the LCAO-ASMO-SCF method differ significantly from the actual measurement results, even when using AOs that are thought to be reliable. Therefore, the Pariser-Parr-Pople (PPP) method is used, which is a semi-empirical method proposed to evaluate atomic integrals using empirical values ​​and measure agreement with experimental values. In this method, the one-center Coulomb integral is defined as the measured ionization potential of the valence state minus the electron affinity. The PPP method is now widely used and has become the standard method for MO calculations.

Source: Morikita Publishing "Chemical Dictionary (2nd Edition)" Information about the Chemical Dictionary 2nd Edition

略称MO法．分子の電子状態を定める近似法の一種．これに対して，原子軌道法(AO法)または原子価結合法(VB法)とよばれる近似法がある．多電子系の分子にシュレーディンガーの波動方程式を適用しても，多体問題や多中心の問題が加わって解くことはできない．そのため，いろいろな近似方法が行われているが，それらの考え方の出発点となっているものがMO法とAO法に大別される．どちらも分子軌道関数は，普通，原子軌道(原子の波動関数)によって組み立てられる．AO法では，電子は構成原子に強く束縛されているという立場から分子を考えるのに対して，MO法では，電子は原子から解放されて分子全体に広がっていると考える．このように，AO法では構成原子の個性を重視するが，MO法ではこれを無視する．
しかし，どちらの方法から出発しても，近似を高めていけばよく一致した結論が得られる．MO法では分子全体に広がっている軌道が，原子に近いところではその原子軌道に似た形となるとして，分子軌道をしばしば構成原子の原子軌道(AO)の線形結合として近似する．これをLCAO近似という．分子軌道(MO)の数は用いたAOの数と等しいが，各MOはパウリの原理によってスピン逆平行の電子を2個まで収容できる．基底状態はエネルギーの低い順にMOに2個ずつ電子を入れることによって与えられる．たとえば，水素分子では結合性分子軌道にスピン逆平行の2個の電子が入って基底状態がつくられる．この場合，電子は分子全体に広がって運動しているので，2個の電子が同時に一方の原子に集まる確率が生じ，イオン構造の寄与が過大に見積もられる結果になる．これを補正するために，ほかの状態(励起状態)を取り入れる方法が用いられる．これを配置間相互作用(CI)の近似という．
LCAO近似のMO法のなかで，CIをあらわに考慮しないもっとも簡単な方法として用いられるものに，ヒュッケル(Hückel)近似がある．この方法の特徴は，まず多電子系のハミルトニアンを，各電子に電子間相互作用を均等にわりあてて，有効な一電子ハミルトニアンの和として表す．次に，このようにして得られる行列要素を見積もるのに，炭化水素のπ電子系の場合，クーロン積分と共鳴積分を経験的なパラメーターとみなし，実測値を再現するように決める．ヒュッケル法は定量的な研究には向かないが種々の分子を比較検討する場合に有効である．
ヒュッケルMO法はπ電子系だけに適用されるが，σ電子系を扱う拡張されたヒュッケル法(extended Hückel method)がある．この方法では，内殻電子(1s電子)を除くすべての原子軌道を用いてLCAO-MOをつくり，クーロン積分は原子価状態のイオン化電位で見積もり，共鳴積分はクーロン積分と重なり積分から計算される．電子間相互作用をあらわに考慮したMO法は反対称化分子軌道(ASMO)法とよばれる．ASMO法では，スピン関数を含めた全系の波動関数がパウリの原理を満足するように反対称化されている．一般に，対称性の高い特別の分子を除いては，LCAO-MOのなかのAOの係数をはじめから知ることはできない．このような場合，ハートリー-フォック(Hartree-Fock)の自己無撞(どう)着場(selfconsistent field，SCF)の方法と同様の手続きでMOが決められている．LCAO-ASMO-SCF法による理論的な計算結果は，信頼できると思われるAOを用いても実測結果とはかなりの差を生じる．そこで，原子積分を経験的な値を用いて評価し，実験値との一致をはかる方法として提案された半経験的な方法であるパリザー-パール-ポープル(Pariser-Parr-Pople，PPP)法が利用される．この方法では，一中心クーロン積分を実測による原子価状態のイオン化電位から電子親和力を引いた値として定める．PPP法が現在広く利用され，MO計算の標準的な方法になっている．

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