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electron orbitals are defined as the region where an electron has a 90% probability to be.

I know this is quite a widespread definition. Unfortunately, it is completely wrong. Let me explain why, and after, I'll discuss their shape.

Definition of orbital

Although the definition of orbitals and even fields as the region of space such... has been used in the history of Physics by famous scientists, it is mathematical and physical nonsense. The reason is simple. An orbital can be represented by positive, negative, and even complex numbers. Do we know negative or complex regions of space? We add the values of overlapping orbitals. Do we add overlapping regions of space?

An orbital is actually a one-particle wavefunction, say $\psi({\bf r})$, in general complex, with the three-dimensional space as its domain. Its relation with probabilities is that the integral of $ |\psi({\bf r})|^2$ over every three-dimensional volume gives the probability of finding an electron in that region of space.

Therefore, at best, the usual definition is missing some important additional words. For example, we could say consistently that, among many different ways of representing an orbital, we can choose to use the boundary surface of the region of space where there is a given percentage of finding one electron (not necessarily $90\%$, another frequently chosen threshold is $95\%$). For example, by selecting the threshold of $90\%$ of probability, we are using as a visual representation of an orbital $\psi({\bf r})$ the surface delimiting the three-dimensional volume $V$ such that $$ \int_V |\psi({\bf r})|^2 d^3{\bf r}= 0.9.\tag{1} $$

Of course, and I think this is your main concern, equation ($1$) does not uniquely identify a region of space. Then, some additional specifications should be added. Actually, the generally used convention is that the encompassed region is the region obtained by considering the three-dimensional volume we get by adding shells of decreasing probability density starting from the absolute maximum.

Notice that this remains a possible way to get an idea of the shape of an orbital, but it does not coincide with the orbital. Surfaces are finite and limited, while orbitals are extended everywhere. Moreover, the definition based on a given probability level implies that one is using the modulus squared of the orbital instead of the orbital itself.

Notice that, due to many different conventions that may be used to provide a graphic representation of an orbital, it is advised to look for a clear definition of the convention used when looking at pictures of orbitals.

Shapes

The shapes of the orbitals are a consequence of the way the wavefunction depends on the electronic coordinates at fixed nuclear positions. They may vary greatly depending on the kind of orbitals (think, for example, the huge variety of molecular orbitals). Probably the simplest cases of orbitals are the hydrogenoid atomic orbitals that are tabulated and graphically represented in many ways.

electron orbitals are defined as the region where an electron has a 90% probability to be.

I know this is quite a widespread definition. Unfortunately, it is completely wrong. Let me explain why, and after, I'll discuss their shape.

Definition of orbital

Although the definition of orbitals and even fields as the region of space such... has been used in the history of Physics by famous scientists, it is mathematical and physical nonsense. The reason is simple. An orbital can be represented by positive, negative, and even complex numbers. Do we know negative or complex regions of space? We add the values of overlapping orbitals. Do we add overlapping regions of space?

An orbital is actually a one-particle wavefunction, say $\psi({\bf r})$, in general complex, with the three-dimensional space as its domain. Its relation with probabilities is that the integral of $ |\psi({\bf r})|^2$ over every three-dimensional volume gives the probability of finding an electron in that region of space.

Therefore, at best, the usual definition is missing some important additional words. For example, we could say consistently that, among many different ways of representing an orbital, we can choose to use the boundary surface of the region of space where there is a given percentage of finding one electron (not necessarily $90\%$, another frequently chosen threshold is $95\%$). For example, by selecting the threshold of $90\%$ of probability, we are using as a visual representation of an orbital $\psi({\bf r})$ the surface delimiting the three-dimensional volume $V$ such that $$ \int_V |\psi({\bf r})|^2 d^3{\bf r}= 0.9.\tag{1} $$

Of course, and I think this is your main concern, equation ($1$) does not uniquely identify a region of space. Then, some additional specifications should be added. Actually, the generally used convention is that the encompassed region is the region obtained by considering the three-dimensional volume we get by adding shells of decreasing probability density starting from the absolute maximum. In this way, the boundary surface of the volume is an iso-$ |\psi({\bf r})|^2 $ surface containing all the points where the square modulus of the wavefunction is higher.

Notice that this remains a possible way to get an idea of the shape of an orbital, but it does not coincide with the orbital. Surfaces are finite and limited, while orbitals are extended everywhere. Moreover, the definition based on a given probability level implies that one is using the modulus squared of the orbital instead of the orbital itself.

Notice that, due to many different conventions that may be used to provide a graphic representation of an orbital, it is advised to look for a clear definition of the convention used when looking at pictures of orbitals.

Shapes

The shapes of the orbitals are a consequence of the way the wavefunction depends on the electronic coordinates at fixed nuclear positions. They may vary greatly depending on the kind of orbitals (think, for example, the huge variety of molecular orbitals). Probably the simplest cases of orbitals are the hydrogenoid atomic orbitals that are tabulated and graphically represented in many ways.

electron orbitals are defined as the region where an electron has a 90% probability to be.

I know this is quite a widespread definition. Unfortunately, it is completely wrong. Let me explain why, and after, I'll discuss their shape.

Definition of orbital

Although the definition of orbitals and even fields as the region of space such... has been used in the history of Physics by famous scientists, it is mathematical and physical nonsense. The reason is simple. An orbital can be represented by positive, negative, and even complex numbers. Do we know negative or complex regions of space? We add the values of overlapping orbitals. Do we add overlapping regions of space?

An orbital is actually a one-particle wavefunction, say $\psi({\bf r})$, in general complex, with the three-dimensional space as its domain. Its relation with probabilities is that the integral of $ |\psi({\bf r})|^2$ over every three-dimensional volume gives the probability of finding an electron in that region of space.

Therefore, at best, the usual definition is missing some important additional words. For example, we could say consistently that, among many different ways of representing an orbital, we can choose to use the boundary surface of the region of space where there is a given percentage of finding one electron. For example, by selecting the threshold of $90\%$ of probability, we are using as a visual representation of an orbital $\psi({\bf r})$ the surface delimiting the three-dimensional volume $V$ such that $$ \int_V |\psi({\bf r})|^2 d^3{\bf r}= 0.9.\tag{1} $$

Of course, and I think this is your main concern, equation ($1$) does not uniquely identify a region of space. Then, some additional specifications should be added. Actually, the encompassed region is the region obtained by considering the three-dimensional volume we get by adding shells of decreasing probability density. Starting from the absolute maximum.

Notice that this remains a possible way to get an idea of the shape of an orbital, but it does not coincide with the orbital. Surfaces are finite and limited, while orbitals are extended everywhere. Moreover, the definition based on a given probability level implies that one is using the modulus squared of the orbital instead of the orbital itself.

Notice that, due to many different conventions that may be used to provide a graphic representation of an orbital, it is advised to look for a clear definition of the convention used when looking at pictures of orbitals.

Shapes

The shapes of the orbitals are a consequence of the way the wavefunction depends on the electronic coordinates at fixed nuclear positions. They may vary greatly depending on the kind of orbitals (think, for example, the huge variety of molecular orbitals). Probably the simplest cases of orbitals are the hydrogenoid atomic orbitals that are tabulated and graphically represented in many ways.

electron orbitals are defined as the region where an electron has a 90% probability to be.

I know this is quite a widespread definition. Unfortunately, it is completely wrong. Let me explain why, and after, I'll discuss their shape.

Definition of orbital

Although the definition of orbitals and even fields as the region of space such... has been used in the history of Physics by famous scientists, it is mathematical and physical nonsense. The reason is simple. An orbital can be represented by positive, negative, and even complex numbers. Do we know negative or complex regions of space? We add the values of overlapping orbitals. Do we add overlapping regions of space?

An orbital is actually a one-particle wavefunction, say $\psi({\bf r})$, in general complex, with the three-dimensional space as its domain. Its relation with probabilities is that the integral of $ |\psi({\bf r})|^2$ over every three-dimensional volume gives the probability of finding an electron in that region of space.

Therefore, at best, the usual definition is missing some important additional words. For example, we could say consistently that, among many different ways of representing an orbital, we can choose to use the boundary surface of the region of space where there is a given percentage of finding one electron (not necessarily $90\%$, another frequently chosen threshold is $95\%$). For example, by selecting the threshold of $90\%$ of probability, we are using as a visual representation of an orbital $\psi({\bf r})$ the surface delimiting the three-dimensional volume $V$ such that $$ \int_V |\psi({\bf r})|^2 d^3{\bf r}= 0.9.\tag{1} $$

Of course, and I think this is your main concern, equation ($1$) does not uniquely identify a region of space. Then, some additional specifications should be added. Actually, the generally used convention is that the encompassed region is the region obtained by considering the three-dimensional volume we get by adding shells of decreasing probability density starting from the absolute maximum.

Notice that this remains a possible way to get an idea of the shape of an orbital, but it does not coincide with the orbital. Surfaces are finite and limited, while orbitals are extended everywhere. Moreover, the definition based on a given probability level implies that one is using the modulus squared of the orbital instead of the orbital itself.

Notice that, due to many different conventions that may be used to provide a graphic representation of an orbital, it is advised to look for a clear definition of the convention used when looking at pictures of orbitals.

Shapes

The shapes of the orbitals are a consequence of the way the wavefunction depends on the electronic coordinates at fixed nuclear positions. They may vary greatly depending on the kind of orbitals (think, for example, the huge variety of molecular orbitals). Probably the simplest cases of orbitals are the hydrogenoid atomic orbitals that are tabulated and graphically represented in many ways.

electron orbitals are defined as the region where an electron has a 90% probability to be.

I know this is quite a widespread definition. Unfortunately, it is completely wrong. Let me explain why, and after, I'll discuss their shape.

Definition of orbital

Although the definition of orbitals and even fields as the region of space such... has been used in the history of Physics by famous scientists, it is mathematical and physical nonsense. The reason is simple. An orbital can be represented by positive, negative, and even complex numbers. Do we know negative or complex regions of space? We add the values of overlapping orbitals. Do we add overlapping regions of space?

An orbital is actually a one-particle wavefunction, say $\psi({\bf r})$, in general complex, with the three-dimensional space as its domain. Its relation with probabilities is that the integral of $ |\psi({\bf r})|^2$ over every three-dimensional volume gives the probability of finding an electron in that region of space.

Therefore, at best, the usual definition is missing some important additional words. For example, we could say consistently that, among many different ways of representing an orbital, we can choose to use the boundary surface of the region of space where there is a given percentage of finding one electron. For example, by selecting the threshold of $90\%$ of probability, we are using as a visual representation of an orbital $\psi({\bf r})$ the surface delimiting the three-dimensional volume $V$ such that $$ \int_V |\psi({\bf r})|^2 d^3{\bf r}= 0.9.\tag{1} $$

Of course, and I think this is your main concern, equation ($1$) does not uniquely identify a region of space. Then, some additional specifications should be added. Actually, the encompassed region is the region obtained by considering the three-dimensional volume we get by adding shells of decreasing probability density. Starting from the absolute maximum.

Notice that this remains a possible way to get an idea of the shape of an orbital, but it does not coincide with the orbital. Surfaces are finite and limited, while orbitals are extended everywhere. Moreover, the definition based on a given probability level implies that one is using the modulus squared of the orbital instead of the orbital itself.

Shapes

The shapes of the orbitals are a consequence of the way the wavefunction depends on the electronic coordinates at fixed nuclear positions. They may vary greatly depending on the kind of orbitals (think, for example, the huge variety of molecular orbitals). Probably the simplest cases of orbitals are the hydrogenoid atomic orbitals that are tabulated and graphically represented in many ways.

electron orbitals are defined as the region where an electron has a 90% probability to be.

I know this is quite a widespread definition. Unfortunately, it is completely wrong. Let me explain why, and after, I'll discuss their shape.

Definition of orbital

Although the definition of orbitals and even fields as the region of space such... has been used in the history of Physics by famous scientists, it is mathematical and physical nonsense. The reason is simple. An orbital can be represented by positive, negative, and even complex numbers. Do we know negative or complex regions of space? We add the values of overlapping orbitals. Do we add overlapping regions of space?

An orbital is actually a one-particle wavefunction, say $\psi({\bf r})$, in general complex, with the three-dimensional space as its domain. Its relation with probabilities is that the integral of $ |\psi({\bf r})|^2$ over every three-dimensional volume gives the probability of finding an electron in that region of space.

Therefore, at best, the usual definition is missing some important additional words. For example, we could say consistently that, among many different ways of representing an orbital, we can choose to use the boundary surface of the region of space where there is a given percentage of finding one electron. For example, by selecting the threshold of $90\%$ of probability, we are using as a visual representation of an orbital $\psi({\bf r})$ the surface delimiting the three-dimensional volume $V$ such that $$ \int_V |\psi({\bf r})|^2 d^3{\bf r}= 0.9.\tag{1} $$

Of course, and I think this is your main concern, equation ($1$) does not uniquely identify a region of space. Then, some additional specifications should be added. Actually, the encompassed region is the region obtained by considering the three-dimensional volume we get by adding shells of decreasing probability density. Starting from the absolute maximum.

Notice that this remains a possible way to get an idea of the shape of an orbital, but it does not coincide with the orbital. Surfaces are finite and limited, while orbitals are extended everywhere. Moreover, the definition based on a given probability level implies that one is using the modulus squared of the orbital instead of the orbital itself.

Notice that, due to many different conventions that may be used to provide a graphic representation of an orbital, it is advised to look for a clear definition of the convention used when looking at pictures of orbitals.

Shapes

The shapes of the orbitals are a consequence of the way the wavefunction depends on the electronic coordinates at fixed nuclear positions. They may vary greatly depending on the kind of orbitals (think, for example, the huge variety of molecular orbitals). Probably the simplest cases of orbitals are the hydrogenoid atomic orbitals that are tabulated and graphically represented in many ways.