Exhibit 4.1


 


NUMBER U-


 


SEE REVERSE FOR CERTAIN DEFINITIONS


 


CUSIP 039562 202


 


ARCHIMEDES
TECH SPAC PARTNERS CO.


 


UNITS CONSISTING OF (I) ONE SUBUNIT,
CONSISTING OF ONE SHARE OF COMMON STOCK


AND ONE-QUARTER OF ONE REDEEMABLE WARRANT,


AND (II) ONE-QUARTER OF ONE REDEEMABLE
WARRANT,


EACH WHOLE WARRANT ENTITLING THE HOLDER
TO PURCHASE ONE SHARE OF COMMON STOCK


 





	 	THIS CERTIFIES THAT	is
the owner of	Units.









 


Each Unit (“Unit”)
consists of (i) one (1) subunit (“Subunits”), consisting of one share of common stock, par value $0.0001 per share
(“Common Stock”), of Archimedes Tech SPAC Partners Co., a Delaware corporation (the “Company”),
and one-quarter of one (1) warrant (“Warrant”), and (ii) one-quarter of one (1) Warrant. Each whole Warrant
entitles the holder to purchase one share of Common Stock for $11.50 per share (subject to adjustment). Each whole Warrant will
become exercisable 30 days after the Company’s completion of an initial merger, capital stock exchange, asset acquisition,
or other similar business combination with one or more businesses or entities (a “Business Combination”), and
will expire unless exercised before 5:00 p.m., New York City Time, on the fifth anniversary of the completion of an initial Business
Combination, or earlier upon redemption or liquidation. The Subunit(s) and Warrant(s) comprising the Unit(s) represented by this
certificate are not transferable separately until ninety days following the Company’s initial public offering, unless EarlyBirdCapital,
Inc. informs the Company of its decision to allow earlier separate trading, except that in no event will the Common Stock and Warrants
be separately tradeable until the Company has filed an audited balance sheet reflecting the Company’s receipt of the gross
proceeds of its initial public offering and issued a press release announcing when such separate trading will begin. The terms
of the Warrants are governed by a Warrant Agreement, dated as of ___________, 2021, between the Company and Continental Stock Transfer
& Trust Company, as Warrant Agent, and are subject to the terms and provisions contained therein, all of which terms and provisions
the holder of this certificate consents to by acceptance hereof. Copies of the Warrant Agreement are on file at the office of the
Warrant Agent at 1 State Street, 30th Floor, New York, New York 10004, and are available to any Warrant holder on written request
and without cost.


 


This certificate is
not valid unless countersigned by the Transfer Agent and Registrar of the Company. Witness the facsimile seal of the Company and
the facsimile signatures of its duly authorized officers.


 


	 	 	 
	Chief Executive Officer	 	Secretary





 


Archimedes Tech SPAC Partners Co.


 


The Company will furnish
without charge to each unitholder who so requests, a statement of the powers, designations, preferences and relative, participating,
optional or other special rights of each class of stock or series thereof of the Company and the qualifications, limitations, or
restrictions of such preferences and/or rights.


 






    


    




    

 


The following abbreviations,
when used in the inscription on the face of this certificate, shall be construed as though they were written out in full according
to applicable laws or regulations:


 


	TEN COM	 	—   	as tenants in common	 	UNIF GIFT MIN ACT —	           Custodian         
	TEN ENT	 	—	as tenants by the entireties	 	 	(Cust)                  (Minor)
	JT TEN	 	—	as joint tenants with right of survivorship and not as tenants in common	 	 	

                     

                    
under Uniform Gifts to Minors
	 	 	 	 	 	 	Act	 
	 	 	 	 	 	 	 	(State)



 








Additional abbreviations may also be used though not in the
above list.


 


For value received,___________hereby sell, assign and transfer
unto___________


 


PLEASE INSERT SOCIAL SECURITY OR OTHER IDENTIFYING NUMBER OF
ASSIGNEE


 


(PLEASE PRINT OR TYPEWRITE NAME AND ADDRESS,
INCLUDING ZIP CODE, OF ASSIGNEE)


  


___________________represented by the within Certificate,
and do hereby irrevocably constitute and appoint


 


__________________to transfer the said Units on the books of
the within named Company with full power of substitution in the premises.


 


Dated: _________________________ 


  





	 	 
	 	Notice:The signature to this assignment must correspond with the name as written upon the face of the certificate in every particular, without alteration or enlargement or any change whatever.


 


Signature(s) Guaranteed:


 


	 	 
	THE SIGNATURE(S) MUST BE GUARANTEED BY AN ELIGIBLE GUARANTOR INSTITUTION (BANKS, STOCKBROKERS, SAVINGS AND LOAN ASSOCIATIONS AND CREDIT UNIONS WITH MEMBERSHIP IN AN APPROVED SIGNATURE GUARANTEE MEDALLION PROGRAM, PURSUANT TO S.E.C. RULE 17Ad-15 (OR ANY SUCCESSOR RULE).	 


 


The holder(s) of this certificate shall
be entitled to receive a pro-rata portion of the funds from the trust account with respect to the Subunits underlying this certificate
only in the event that (i) the Corporation is forced to liquidate because it does not consummate an initial business combination
within the period of time set forth in the Corporation’s Amended and Restated Certificate of Incorporation, as the same may
be amended from time to time (the “Charter”) or (ii) if the holder seeks to convert his subunits upon consummation
of, or sell his subunits in a tender offer in connection with, an initial business combination or in connection with certain amendments
to the Charter. In no other circumstances shall the holder(s) have any right or interest of any kind in or to the trust account
with respect to the subunit underlying the unit.