My spouse and I will maintain separate households. We will not live together due to children. Under these circumstances, can we each file Head of Household?

Theoretically, you can.

https://ttlc.intuit.com/questions/3692749-if-married-in-october-but-don-t-live-together-can-we-file-head-of-household

https://ttlc.intuit.com/questions/2567672-can-a-married-person-claim-head-of-household-filing-status

Not only must you each have a qualifying person (usually a child) living with you, you must not  have spent even one night in the other spouse's home during the last half of the year. http://finance.zacks.com/can-married-woman-head-household-irs-6373.html

Even if you can, you should do comparative returns as Married Filing Jointly may still be better. You can use this tool  https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1

 Probably not, and it would be very dangerous to try. 

 First of all, you can't claim head of household status unless you provide care in your home for a qualifying dependent, usually your child, who must live with you for at least half of the year. So if you have children with another parent, you have to be the custodial parent in order to claim the child as a qualified person for head of household. 

 Then, if you are married, you cannot use the head of household status unless you are separated for at least all of the last six months of the tax year.  You can't live in the same home or share the same household. A household is not necessarily a house, or apartment, or particular structure. For example, two families who share a house to save on rent, can be two different households if they lead essentially separate lives. On the other hand, one family that has two homes, such as a main home and a vacation home, cannot claim to be two separate households, even if the two properties are in different peoples names.  They are one household, because they share finances, they share activities, they share meals, they shared decisions on raising the children, they share clothing shopping, and everything else that makes them a family.

 I think it is very unlikely that if you are married, you would be two separate households in fact, even if you live at two different addresses. And I think the IRS would probably view it as a sham to get favorable tax treatment.   You would have to work very hard to convince the IRS that you were existing in two completely separate households in the full meaning of the word, rather than simply dividing your time between two properties who's ownership is in different names. 

I don't want to claim head of household