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Walsh Diagram For Tri And Penta Atomic Molecules Pdf 98


Walsh diagram for tri and penta atomic molecules




A Walsh diagram is a graphical representation of the calculated orbital energies of a molecule as a function of a geometrical distortion parameter, such as bond angle or bond length. Walsh diagrams are useful for making quick predictions about the shapes and stabilities of small molecules, as well as their electronic transitions and spectra.


In this article, we will focus on the Walsh diagrams for triatomic and penta atomic molecules, which are often called angular correlation diagrams or correlation diagrams. These diagrams show how the molecular orbitals change as the bond angle between the central atom and the two terminal atoms varies from 180 degrees (linear) to 90 degrees (bent). We will use the linear combination of atomic orbitals (LCAO) method to construct the molecular orbitals from the available atomic orbitals on each atom.


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Triatomic molecules




A general triatomic molecule can be written as AH2, where A is a second-row element and H is hydrogen. The molecular orbitals can be expressed as a linear combination of the 1s orbitals on the two hydrogens (H1 and H2) and the four n=2 orbitals (2s, 2px, 2py, 2pz) on the central atom A:


χ⟩ = a11s⟩H1 + a21s⟩H2 + a32s⟩A + a42px⟩A + a52py⟩A + a62pz⟩A


The coefficients of this expansion (ai) are determined by solving the secular determinant. If we consider only linear AH2 molecules, then we can simplify this expression by ignoring the 2px and 2py atomic orbitals since they are perpendicular to the bonds and are hence non-bonding. Moreover, by symmetry, the coefficients in front of the 1s orbitals of hydrogen are equal. Therefore, we have:


χ⟩ = a11s⟩H1 + a11s⟩H2 + a32s⟩A + a62pz⟩A


We can construct four molecular orbitals from these four atomic orbitals: one σ orbital, one σ* orbital, and two non-bonding orbitals. The σ orbital is bonding, meaning that it lowers the energy of the molecule compared to the separated atoms. The σ* orbital is antibonding, meaning that it raises the energy of the molecule compared to the separated atoms. The non-bonding orbitals have no effect on the bond strength or length.


The energy of each molecular orbital depends on the overlap between the atomic orbitals, which in turn depends on the bond angle. As the bond angle decreases from 180 degrees to 90 degrees, the overlap between the 1s orbitals of hydrogen increases, while the overlap between the 2s and 2pz orbitals of A decreases. This means that the σ orbital becomes more stable (lower in energy), while the σ* orbital becomes less stable (higher in energy). The non-bonding orbitals remain unchanged in energy.


A Walsh diagram for a triatomic molecule shows these energy changes as curves plotted against the bond angle. For example, Figure 1 shows the Walsh diagram for BeH, which has four valence electrons. The lowest energy configuration is achieved when the bond angle is 180 degrees, which corresponds to a linear shape. This is because the two electrons occupy the σ orbital, which is most stable at this angle. If the bond angle decreases, the energy of the molecule increases due to the destabilization of the σ orbital and the repulsion between the electrons.



Figure 1: Walsh diagram for BeH. The horizontal axis shows the bond angle in degrees, and the vertical axis shows the energy in arbitrary units. The molecular orbitals are labeled as σ, σ*, and n. The dashed line indicates the energy of the separated atoms. The red dots indicate the occupancy of the molecular orbitals by electrons.


Penta atomic molecules




A general penta atomic molecule can be written as AX4, where A is a second-row element and X is a halogen. The molecular orbitals can be expressed as a linear combination of the 2s and 2p orbitals on the central atom A and the 2p orbitals on the four terminal atoms X1, X2, X3, and X4:


χ⟩ = a12s⟩A + a22px⟩A + a32py⟩A + a42pz⟩A + b12p⟩X1 + b22p⟩X2 + b32p⟩X3 + b42p⟩X4


The coefficients of this expansion (ai and bi) are determined by solving the secular determinant. If we consider only tetrahedral AX4 molecules, then we can simplify this expression by using hybrid orbitals on the central atom A. Hybrid orbitals are linear combinations of atomic orbitals that have the same symmetry and shape as the bonds. For a tetrahedral molecule, we can use four sp hybrid orbitals, which are formed by mixing one 2s orbital and three 2p orbitals:


sp⟩A = (2s⟩+2p⟩+2p⟩+2p⟩) / 2 sp⟩A = (2s⟩-2p⟩-2p⟩+2p⟩) / 2 sp⟩A = (2s⟩+2p⟩-2p⟩-2p⟩) / 2 sp⟩ = (_2s_>-_2p_>+_2p_>+_2p_>) / 2 The molecular orbitals can then be expressed as a linear combination of these four sp^3 hybrid orbitals on A and the four 2p orbitals on X: χ⟩ = c1sp⟩A + c2sp⟩A + c3sp⟩A + c4sp⟩ + d12p⟩X1 + d22p⟩X2 + d32p⟩X3 + d42p⟩X4


The coefficients of this expansion (ci and di) are determined by solving the secular determinant. We can construct eight molecular orbitals from these eight atomic orbitals: four σ orbitals, two π orbitals, and two π* orbitals. The σ orbitals are bonding or non-bonding, depending on the sign of the coefficients. The π orbitals are bonding, and the π* orbitals are antibonding. The energy of each molecular orbital depends on the overlap between the hybrid orbitals and the 2p orbitals, which in turn depends on the bond angle.


A Walsh diagram for a penta atomic molecule shows these energy changes as curves plotted against the bond angle. For example, Figure 2 shows the Walsh diagram for CH4, which has eight valence electrons. The lowest energy configuration is achieved when the bond angle is 109.5 degrees, which corresponds to a tetrahedral shape. This is because the eight electrons occupy the four σ orbitals, which are most stable at this angle. If the bond angle increases or decreases, the energy of the molecule increases due to the destabilization of the σ orbitals and the repulsion between the electrons.


Figure 2: Walsh diagram for CH4. The horizontal axis shows the bond angle in degrees, and the vertical axis shows the energy in arbitrary units. The molecular orbitals are labeled as σ, π, and π*. The dashed line indicates the energy of the separated atoms. The red dots indicate the occupancy of the molecular orbitals by electrons.


I hope this article helps you understand the concept of Walsh diagrams for tri and penta atomic molecules. If you have any questions or feedback, please let me know.


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