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**Lecture 1 THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM**

1) Theoretical description of the hydrogen atom: Schrödinger equation: HY = EY H H is the Hamiltonian operator (kinetic + potential energy of an electron) E – the energy of an electron whose distribution in space is described by Y - wave function YY* is proportional to the probability of finding the electron in a given point of space If Y is normalized, the following holds: The part of space where the probability to find electron is non-zero is called orbital When we plot orbitals, we usually show the smallest volume of space where this probability is 90%. Comments on the Schrodinger equation:

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**2) Solution of the Schrödinger equation for the hydrogen atom**

2) Solution of the Schrödinger equation for the hydrogen atom. Separation of variables r, q, f For the case of the hydrogen atom the Schrödinger equation can be solved exactly when variables r, q, f are separated like Y = R(r) Q(q) F(f) Three integer parameters appear in the solution, n (principal quantum number), while solving R-equation l (orbital quantum number), while solving Q-equation ml (magnetic quantum number), while solving F-equation n = 1, 2, … ∞; defines energy of an electron l = 0, … n-2, n-1 (total n values); defines shape of an orbital: 0-s, 1-p, 2-d, 3-f ,... ml = -l, -l+1, …, 0, …, l-1, l (total 2l+1 values); defines spatial orientation The number of all possible combinations of l and ml for a given n is Solution of the Schrodinger equation for the case of the hydrogen atom:

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**Plot of the hydrogen atom orbitals**

s orbitals p orbitals d orbitals

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**3) The hydrogen atom orbitals**

Radial components of the hydrogen atom wavefunctions look as follows: R(r) = for n = 1, l = 0 1s orbital R(r) = for n = 2, l = 0 2s orbital R(r) = for n = 2, l = 1, ml = 0 2pz orbital R(r) = n = 3, l = 0 3s orbital R(r) = n = 3, l = 1, ml = 0 3pz orbital Here a0 = , the first orbit radius (0.529Å). Z is the nuclear charge (+1)

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**4) Radial components of the hydrogen atom orbitals**

The exponential decay of R(r) is slower for greater n At some distances r where R(r) is equal to 0, we have radial nodes, n-l-1 in total 1s 2s 2p node

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**5) Radial probability function**

Radial probability function is defined as [r R(r)]2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: 1s 2p 2s

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**Angular components of the hydrogen atom wave functions, QF **

for l = 0, ml = 0; QF = s orbitals for l = 1, ml = 0; QF = pz orbitals for l = 2, ml = 0; QF = dz2 orbitals For each orbital we have l angular nodal surfaces Therefore, together with radial nodes we have in total (n-l-1)+l = n-1 nodes for each orbital Hydrogen-like orbitals (animated):

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**7) Plot of the hydrogen atom orbitals**

Orbitals are called gerade (g) if they have center of symmetry or ungerade (u) if they do not have it Examples of orbitals which are gerade: s, d. Orbitals which are ungerade: p.

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**8) Allowed energies of an electron in the hydrogen atom**

Note: E does not depend on either l or ml In other words, for each given n s, p, d etc orbitals of the hydrogen atom are of the same energy (degenerate) Spectrum of hydrogen atom: 4s, 4p, 4d, 4f 3s, 3p, 3d 2s, 2px, 2py, 2pz 1s

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Summary Exact energy and spatial distribution of an electron in the hydrogen atom can be found by solving Shrödinger equation; Three quantum numbers n, l, ml appear as integer parameters while solving the equation; Each hydrogen orbital can be characterized by energy (n), shape (l), spatial orientation (ml), number of nodal surfaces (n-1) and symmetry (g, u).

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Chapter 7 The Quantum-Mechanical Model of the Atom

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