Which elements have the quantum numbers (4,3,-2,+1/2) and (5,2,-1,-1/2)

The elements are Samarium (Sm) and Silver (Ag).

There are four quantum numbers which are used to describe the distribution of electrons in an atom;

For the first element with quantum numbers (4,3,-2,+1/2);

Principal quantum number n = 4, Azimuthal quantum number, ℓ = 3, Magnetic quantum number, m = -2 and Spin quantum number, s =  +½

The element belongs to period 4, has 4f electrons, has electrons in the -2 or sixth sub-orbital spinning in the +½ direction. Therefore, the element is Samarium (Sm).

For the second element with quantum numbers (5,2,-1,-1/2);

Principal quantum number n = 5, Azimuthal quantum number, ℓ = 2, Magnetic quantum number, m = -1 and Spin quantum number, s =  -½

The element belongs to period 5, has 4d electrons, has electrons in the -1 or fourth sub-orbital spinning in the -½ direction. Therefore, the element is Silver (Ag).

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The elements are samarium (Sm) and silver (Ag).  

Quantum numbers (4,3,-2,+½)  

n = 4: Principal quantum number = 4.  

l = 3: Element has 4f electrons  

It is conventional to list quantum numbers in decreasing order.  

Hund’s rule states that all the orbitals must be half-filled with electrons having the same spin before any can receive a second electron.  

We get the following table:  

Element n l   mₗ mₛ  

    La      4 3   3 +½  

    Ce     4 3   2 +½  

    Pr      4 3   1  +½  

    Nd    4 3   0 +½  

    Pm   4 3  -1   +½  

    Sm   4 3 -2 +½  

The element is samarium, Sm.  

Quantum numbers (5,2,-1,-½)  

n = 5: Principal quantum number = 5.  

l = 2: Element has 5d electrons  

We get the following table:  

Element n  l mₗ mₛ  

    Y       5 2  2 +½  

    Zr      5 2  1 +½  

    Nb    5 2  0 +½  

    Mo   5 2  -1 +½  

    Tc    5 2  -2 +½  

    Ru   5 2   2  -½  

    Rh   5 2   1   -½  

    Pd   5 2   0  -½  

    Ag   5 2  -1  -½  

The element is silver, Ag.

Note: These assignments assume that there are no exceptions in the Periodic Table.