What are the atomic orbitals and how are they derived?
user
Sure! I'd be happy to explain atomic orbitals and how they're derived.

Atomic orbitals are mathematical functions that describe the spatial distribution of the electron cloud around an atom in a molecule. They relate to the electron's probability of being found at a particular point in space.

Mathematically, orbital forms depend on the quantum numbers, which determine the type of orbital. These quantum numbers consist of:

1. **Principal quantum number (n)**: Indicates the energy level the electron occupies. Generally, it is given integers in a range of [1, ∞].

2. **Angular momentum quantum number (ℓ)**: Equals the total angular momentum of the electron, including all interactions with the nucleus. It is given integers less than or equal to (n - 1).

3. **Magnetic quantum number (mℓ)**: Denotes how much of the orbital's angular momentum is along a set framework (z-axis) identified an orbital. Given a few less than or corresponding to ℓ.

4. **Spin quantum number (m drew)**: Represents the spin of the atom (arrow-shaped projection of the electron's rotation direction within the plane made by two equal perpendicular momentum vectors). It can have only two possible values:  + 1⁄2 for α spin or −1⁄2 for β spin.

Furthermore, a series of equations from quantum mechanics is used to deduce orbital formulas, primarily the Schrödinger equation (Hψ = Eψ). As a solution to this differential formula, with minor conditions and stringent restrictions, one can derive the characteristic (μs, energy, angular-momentum quantum number, etc.) solutions known as orbital wavefunctions.

These solutions, the orbital wavefunctions, display a complex oval-shaped sphere surrounding nucleus. Such waveform forms depict where the electron is most likely to be situated. Still, not precisely where it will be often, since it might be 50 million miles maybe closer, etc.
assistant
What are the atomic orbitals and how are they derived?
user
Atomic orbitals are mathematical functions that describe the behavior of electrons in an atom. They provide information about the probability of finding an electron in a particular region around the nucleus of an atom. The concept of atomic orbitals is fundamental to the quantum mechanical model of the atom, which supersedes the earlier Bohr model.

Here's a simplified explanation of how they are derived:

1. **Schrödinger Equation**: The basis for deriving atomic orbitals is the Schrödinger equation, a fundamental equation in quantum mechanics. This equation describes how the quantum state of a quantum system changes over time.

2. **Wave Functions**: Solving the Schrödinger equation for an electron in an atom gives wave functions known as orbitals. These wave functions have complex values and can be positive or negative. The square of the wave function at a given point gives the probability density of finding the electron at that point.

3. **Quantum Numbers**: The solutions to the Schrödinger equation depend on three quantum numbers: n (principal quantum number), l (angular momentum quantum number), and m (magnetic quantum number). These numbers define the shape, size, and orientation of the orbital.

4. **Types of Orbitals**: Depending on the value of the angular momentum quantum number (l), orbitals can be of different types: s (l = 0), p (l = 1), d (l = 2), f (l = 3), and so on. Each type has a unique shape: s orbitals are spherical, p orbitals are dumbbell-shaped, d orbitals are more complex and have four lobes, and f orbitals have even more complex shapes.

5. **Pauli Exclusion Principle**: This principle states that no two electrons in an atom can have the same set of four quantum numbers (n, l, m, and s, where s is the spin quantum number). This helps to explain the arrangement of electrons in different orbitals of an atom.
assistant